

DECLASSIFIED 
By authority Secretary of 

SEP 7 1960 

Defense memo 2 August 1960 
LIBRARY OF CONGRESS 



* V* 


DECLASSIFIED 
Dy authority Secretary of 

SEP 7 1960 

Defense memo 2 August 1960 
LIBRARY OF CONGRESS 

SUMMARY TECHNICAL REPORT 
OF THE 

NATIONAL DEFENSE RESEARCH COMMITTEE 


IX? REGULAT ION : BEFORE SERVIC" 
OR REPRODUCING ANY PART OF r 
DOCUMENT, A LL C LASSIFICATION 
MARKINGS MUST EE CANCELLED: 


This document contains information affecting the national defense 
of the United States within the meaning of the Espionage Act, 50 
U. S. C., 31 and 32, as amended. Its transmission or the revelation of 
its contents in any manner to an unauthorized person is prohibited 
by law. 

This volume is classified CONFIDENTIAL in accordance with 
security regulations of the War and Navy Departments because 
certain chapters contain material which was CONFIDENTIAL at 
the date of printing. Other chapters may have had a lower classi- 
fication or none. The reader is advised to consult the War and Navy 
agencies listed on the reverse of this page for the current classifica- 
tion of any material. 


Manuscript and illustrations for this volume were prepared for 
publication by the Summary Reports Group of the Columbia 
University Division of War Research under contract OEMsr-1131 
with the Office of Scientific Research and Development. This vol- 
ume was printed and bound by the Columbia University Press. 

Distribution of the Summary Technical Report of NDRC has 
been made by the War and Navy Departments. Inquiries concern- 
ing the availability and distribution of the Summary Technical 
Report volumes and microfilmed and other reference material 
should be addressed to the War Department Library, Room 
1A-522, The Pentagon, Washington 25, D. C., or to the Office of 
Naval Research, Navy Department, Attention: Reports and 
Documents Section, Washington 25, D. C. 

Copy No. 

78 


This volume, like the seventy others of the Summary Technical 
Report of NDRC, has been written, edited, and printed under 
great pressure. Inevitably there are errors which have slipped past 
Division readers and proofreaders. There may be errors of fact not 
known at time of printing. The author has not been able to follow 
through his writing to the final page proof. 

Please report errors to: 

JOINT RESEARCH AND DEVELOPMENT BOARD 
PROGRAMS DIVISION (STR ERRATA) 

WASHINGTON 25, D. C. 

A master errata sheet will be compiled from these reports and sent 
to recipients of the volume. Your help will make this book more 
useful to other readers and will be of great value in preparing any 
revisions. 


CONFIDENTIAL 


SUMMARY TECHNICAL REPORT OF DIVISION 6, NDRC 


METHODS O Fr>ef3nse 


VOLUME 2A 


_DE CLASSIFIED n 
By authority Secretary of 

SEP 7 1960 

Defense memo 2 August I960 


LIBRARY OF CONGRESS 


OPERATIONS RESEARCH 



OFFICE OF SCIENTIFIC RESEARCH AND DEVELOPMENT 
VANNEVAR BUSH, DIRECTOR 

NATIONAL DEFENSE RESEARCH COMMITTEE 
JAMES B. CONANT, CHAIRMAN 

DIVISION 6 
JOHN T. TATE, CHIEF 


WASHINGTON, D. C., 1946 



C.n. 0 . . (Ktjw r r*3 ffA dJ.xo 


NATIONAL DEFENSE RESEARCH COMMITTEE 

James B. Conant, Chairman 
Richard C. Tolman, Vice Chairman 
Roger Adams Army Representative 1 

Frank B. Jewett Navy Representative 2 

Karl T. Compton Commissioner of Patents 3 

Irvin Stewart, Executive Secretary 


1 Army representatives in order of service: 


Maj. Gen. G. V. Strong 
Maj. Gen. R. C. Moore 
Maj. Gen. C. C. Williams 
Brig. Gen. W. A. Wood, Jr. 

Col. E. 


Col. L. A. Denson 
Col. P. R. Faymonville 
Brig. Gen. E. A. Regnier 
Col. M. M. Irvine 
Routheau 


2 Navy represen tatives in order of service: 

Rear Adm. H. G. Bowen Rear Adm. J. A. Furer 

Capt. Lybrand P. Smith Rear Adm. A. H. Van Keuren 

Commodore FI. A. Schade 
3 Commissioners of Patents in order of sendee: 
Conway P. Coe Casper W. Ooms 


NOTES ON THE ORGANIZATION OF NDRC 


The duties of the National Defense Research Com- 
mittee were (1) to recommend to the Director of 
OSRD suitable projects and research programs on the 
instrumentalities of warfare, together with contract 
facilities for carrying out these projects and pro- 
grams, and (2) to administer the technical and scienti- 
fic work of the contracts. More specifically, NDRC 
functioned by initiating research projects on requests 
from the Army or the Navy, or on requests from an 
allied government transmitted through the Liaison 
Office of OSRD, or on its own considered initiative 
as a result of the experience of its members. Pro- 
posals prepared by the Division, Panel, or Committee 
for research contracts for performance of the work 
involved in such projects were first reviewed by 
NDRC, and if approved, recommended to the Direc- 
tor of OSRD. Upon approval of a proposal by the 
Director, a contract permitting maximum flexibility 
of scientific effort was arranged. The business aspects 
of the contract, including such matters as materials, 
clearances, vouchers, patents, priorities, legal mat- 
ters, and administration of patent matters were 
handled by the Executive Secretary of OSRD. 

Originally NDRC administered its work through 
five divisions, each headed by one of the NDRC 
members. These were: 

Division A— Armor and Ordnance 
Division B— Bombs, Fuels, Gases 8c Chemical Prob- 
lems 

Division C— Communication and Transportation 
Division D— Detection, Controls, and Instruments 
Division E— Patents and Inventions 


In a reorganization in the fall of 1942, twenty-three 
administrative divisions, panels, or committees were 
created, each with a chief selected on the basis of his 
outstanding work in the particular field. The NDRC 
members then became a reviewing and advisory 
group to the Director of OSRD. The final organiza- 
tion was as follows: 

Division 1— Ballistic Research 

Division 2— Effects of Impact and Explosion 

Division 3— Rocket Ordnance 

Division 4— Ordnance Accessories 

Division 5— New Missiles 

Division 6— Sub-Surface Warfare 

Division 7— Fire Control 

Division 8— Explosives 

Division 9— Chemistry 

Division 10— Absorbents and Aerosols 

Division 11— Chemical Engineering 

Division 12— Transportation 

Division 13— Electrical Communication 

Division 14— Radar 

Division 15— Radio Coordination 

Division 16— Optics and Camouflage 

Division 17— Physics 

Division 18— War Metallurgy 

Division 19— Miscellaneous 

Applied Mathematics Panel 

Applied Psychology Panel 

Committee on Propagation 

Tropical Deter’ — Administrative Committee 


Library of Congress 



201 5 


490946 


DECLASSIFIED 
By authority Secretary of 


NDRC FOREWORD SEP 7 I960 


A s events of the years preceding 1940 revealed 
1\. more and more clearly the seriousness of the 
world situation, many scientists in this country came 
to realize the need of organizing scientific research 
for service in a national emergency. Recommenda- 
tions which they made to the White House were given 
careful and sympathetic attention, and as a result 
the National Defense Research Committee [NDRC] 
was formed by Executive Order of the President in 
the summer of 1940. The members of NDRC, ap- 
pointed by the President, were instructed to supple- 
ment the work of the Army and the Navy in the 
development of the instrumentalities of war. A year 
later, upon the establishment of the Office of Scien- 
tific Research and Development [OSRD], NDRC 
became one of its units. 

The Summary Technical Report of NDRC is a 
conscientious effort on the part of NDRC to sum- 
marize and evaluate its work and to present it in a 
useful and permanent form. It comprises some 
seventy volumes broken into groups corresponding 
to the NDRC Divisions, Panels, and Committees. 

The Summary Technical Report of each Division, 
Panel, or Committee is an integral survey of the work 
of that group. The first volume of each group’s re- 
port contains a summary of the report, stating the 
problems presented and the philosophy of attacking 
them and summarizing the results of the research, 
development, and training activities undertaken. 
Some volumes may be “state of the art” treatises 
covering subjects to which various research groups 
have contributed information. Others may contain 
descriptions of devices developed in the laboratories. 
A master index of all these divisional, panel, and 
committee reports which together constitute the 
Summary Technical Report of NDRC is contained 
in a separate volume, which also includes the index 
of a microfilm record of pertinent technical labora- 
tory reports and reference material. 

Some of the NDRC-sponsored researches which 
had been declassified by the end of 1945 were of suffi- 
cient popular interest that it was found desirable to 
report them in the form of monographs, such as the 
series on radar by Division 14 and the monograph on 
sampling inspection by the Applied Mathematics 
Panel. Since the material treated in them is not dupli- 


cated in the 
the monographs are 


of these aspects of NDRC research. 

In contrast to the information on radar, which is of 
widespread interest and much of which is released to 
the public, the research on subsurface warfare is 
largely classified and is of general interest to a more 
restricted group. As a consequence, the report of 
Division 6 is found almost entirely in its Summary 
Technical Report, which runs to over twenty vol- 
umes. The extent of the work of a Division cannot 
therefore be judged solely by the number of volumes 
devoted to it in the Summary Technical Report of 
NDRC: account must be taken of the monographs 
and available reports published elsewhere. 

Any great cooperative endeavor must stand or fall 
with the will and integrity of the men engaged in it. 
This fact held true for NDRC from its inception, 
and for Division 6 under the leadership of Dr. John 
T. Tate. To Dr. Tate and the men who worked with 
him — some as members of Division 6, some as rep- 
resentatives of the Division’s contractors — belongs 
the sincere gratitude of the Nation for a difficult and 
often dangerous job well done. Their efforts contrib- 
uted significantly to the outcome of our naval 
operations during the war and richly deserved the 
warm response they received from the Navy. In 
addition, their contributions to the knowledge of the 
ocean and to the art of oceanographic research will 
assuredly speed peacetime investigations in this field 
and bring rich benefits to all mankind. 

The Summary Technical Report of Division 6, 
prepared under the direction of the Division Chief 
and authorized by him for publication, not only pre- 
sents the methods and results of widely varied re- 
search and development programs but is essentially 
a record of the unstinted loyal cooperation of able 
men linked in a common effort to contribute to the 
defense of their Nation. To them all we extend our 
deep appreciation. 


Vannevar Bush, Director 
Office of Scientific Research and Development 

J. B. Conant, Chairman 
National Defense Research Committee 



FOREWORD 


T his report, constituting Volume 2A of the Di- 
vision 6 Summary Technical Report, describes 
the methods developed and employed during the 
period 1942-1945 by members of the Operations 
Evaluation Group [OEG] in the Headquarters of the 
Commander-in-Chief, U.S. Navy. This group, com- 
posed of civilian scientists, mathematicians, and sta- 
tisticians, and initially designated the Anti-Subma- 
rine Warfare Operations Research Group [ASWORG] 
was established early in 1942 under the supervision 
of the Atlantic Fleet antisubmarine warfare officer, 
Captain (later Rear Admiral) W. D. Baker. 

When Captain Baker requested the formation of 
this group, he stated that the antisubmarine warfare 
unit, as then constituted, was not in a position to 
evaluate properly (1) the probable effectiveness of 
suggested new weapons and (2) developments and 
procedures based upon mathematical studies and 
other available records. 

Captain Baker defined what was desired and in- 
tended by quoting from a RAF Coastal Command 
memorandum on the subject, as follows: 

“Experience over many parts of our war effort has 
shown that such analysis can be of the utmost value, 
and the lack of such analysis can be disastrous. Prob- 
ably the main reason why this is so, is that very many 
war operations involve considerations with which 
scientists are specially trained to compete and in 
which serving officers are in general not trained. This 
is especially the case with all those aspects of opera- 
tions into which probability considerations and the 
theory of errors enter. Serving officers of the highest 
caliber are necessarily employed in important execu- 
tive posts, and are therefore, not available for de- 
tailed analytic work.” 

As to the type of personnel required to staff these 
operations, Captain Baker quoted from the same 
memorandum : 

“A considerable fraction of the staff of an opera- 
tional research section should be of the very highest 
standing in science, and many of them should be 
drawn from those who have had experience at the 
Service Technical Establishments. Others should be 
chosen for analytic ability, e.g., gifted mathemati- 
cians, lawyers, chess players, etc. An ORS which 
contents itself with the routine production of statis- 
tical reports and narratives will be of very limited 
value. The atmosphere required is that of a first class 


pure scientific research institution, and the caliber of 
the personnel should match this. All members of an 
ORS should spend part of their time at operational 
stations in close touch with the flying personnel, and 
where possible, should occasionally go on operational 
or training flights.” 

That the project should be undertaken and the pro- 
posed group established in the Navy was never ques- 
tioned. The experience of the British and of large 
American industries in carrying on very similar 
operational analyses made certain that a suitably 
staffed group could be very effective. The project was 
assigned to Section C-4, NDRC, and under its con- 
tract with Columbia University, a staff was recruited. 

Professor P. M. Morse, of the Massachusetts In- 
stitute of Technology, became project supervisor, 
and Dr. William Shackley was granted a leave of 
absence from the Bell Telephone Laboratories to act 
as director of research. 

The operations performed by the ASWORG 
closely followed Captain Baker’s conception of how 
the group could assist the Navy. Also, Professor 
Morse was able to recruit personnel with qualifica- 
tions very similar to those specified by Captain 
Baker. Although organizational changes occurred, 
the objectives of the group itself were not altered. 
With the creation of the Tenth Fleet as a means of 
coordinating antisubmarine warfare, the group, now 
under the sponsorship of the Office of Field Service 
[OFS], was made an integral part of the Fleet staff. 
As the activities of the group expanded to cover other 
fields, the name was shortened to Operations Re- 
search Group [ORG]. At the end of the war, the 
group was reorganized as the Operations Evaluation 
Group and assigned to the staff of the Chief of Naval 
Operations under a direct Navy contract with the 
Massachusetts Institute of Technology. It is by this 
latest organization that the present report has been 
prepared. 

The material presented in this volume has been 
compiled from reports and memoranda issued by the 
group and from the first hand knowledge and expe- 
rience which the authors gained during the war with 
the techniques and problems discussed. Specific ex- 
amples of these general methods in their application 
to problems of submarine warfare may be found in 
Search and Screening , Volume 2B. These methods also 
underly the more historical treatment given in Vol- 


vii 


CONFIDENTIAL 


FOREWORD 


viii 


ume 3, ASW Operations in World W ar II, as well as 
a wide category of similar investigations. Although the 
procedures employed were designed for applications 
involving the instruments, weapons and conditions 
prevailing during the last war, the methods and sys- 
tematic processes of analysis developed have wide 
application to many military and civilian problems. 

The Division wishes to thank the authors, Pro- 
fessor Morse and Dr. Kimball, for their work in pre- 
paring the volume. It is impossible to name the long 
list of Navy officers and some Army officers who gave 


support to the work of the Operations Research 
Group and who share credit for its performance. The 
full list would include many officers at operational 
bases and stations. However, special acknowledge- 
ment should be made of the cooperation and assis- 
tance received from Admiral W. D. Baker, from 
Admiral F. S. Low, Chief of Staff, Tenth Fleet, of the 
U. S. Navy, and from the British operations research 
groups. 

John T. Tate 
Chief, Division 6 


CONFIDENTIAL 


PREFACE 


I n a sense, ‘this book should have no authors’ 
names, or else several pages full of names. Parts 
of the book were written by various persons during 
the past five years. What the undersigned have done 
is to collect the material, rewrite some in the light of 
later knowledge, expand some to make it intelligible 
for the lay reader, and cement the mosaic into what is 
fondly hoped to be a logical structure. Operations 
research during the war had to be anonymous, and 
much of the work must remain so. 

Since the undersigned were members of the Opera- 
tions Research Group, U. S. Navy, it is perhaps not 
surprising that the examples given are drawn chiefly 
from the work of this group, though an effort has 
been made to include examples from the work of 
other groups. Many persons have helped by discus- 


sions and editorial criticism, including members of 
other operations research groups in this country and 
in England. To mention a few would slight many 
others, so none will be named. 

During the war the scope, methods, and triumphs 
of operations research were not appreciated by most 
scientists or by most military men because no in- 
formation was freely available. If we are not to lose 
this valuable experience and background, some of it 
must be made available to the scientists and engineers 
as well as to the armed services. This is particularly 
important if the methods of operations research have 
important peacetime applications — as it is believed 
they do. 

Philip M. Morse 
George E. Kimball 


CONFIDENTIAL 


IX 





CONTENTS 


CHAPTER PAGE 

1 Introduction 1 

2 Probability 11 

3 The Use of Measures of Effectiveness 38 

4 Strategical Kinematics 61 

5 Tactical Analysis 81 

6 Gunnery and Bombardment Problems 110 

7 Operational Experiments with Equipment and Tactics . .129 

8 Organizational and Procedural Problems 137 

Tables 153 

Bibliography 161 

Contract Numbers 162 

Service Project Numbers 163 

Index 165 


CONFIDENTIAL xi 








Chapter 1 

INTRODUCTION 


I n world war ii, the phrase “operations re- 
search” has come to describe the scientific, quan- 
titative study of operations of war. Perhaps a more 
descriptive term would be “polemology,” from the 
Greek word “polemos” meaning warfare, but the 
more familiar “operations research” will be retained 
in the present treatise because it suggests that the 
methods can be used to study other operations than 
those of war. 

The phrase, operations research, is of recent origin, 
and represents one phase of the expansion of science 
to study more and more complex phenomena. Pre- 
vious aid which science had given warfare was chiefly 
in the direction of providing new weapons and instru- 
ments. Perhaps the most celebrated classical example 
was the work of Archimedes in defending the city of 
Syracuse from the Romans. In the Renaissance, 
Leonardo da Vinci spent much of his engineering 
abilities in military applications, and, in the seven- 
teenth century, the famous Vauban utilized his con- 
siderable geometrical abilities in designing fortifica- 
tions, and his inventive genius in devising the bayonet 
and the method of ricochet fire. Fourier, Monge, and 
Bertholet were retained by Napoleon as his scientific 
advisers. Of these, only Vauban made any real contri- 
butions in applying science to suggest better ways of 
using weapons, rather than adding new “gadgets.” 

In more recent times, warfare has become so com- 
plex that most scientists have confined their contri- 
butions to the devising of new weapons, and have left 
the tactical use of these weapons to the professional 
soldier, and their strategic use to the professional 
politician. The utility, however, of the coordinated 
scientific attack on a wide variety of nonmilitary 
problems has been amply demonstrated in the past 
three decades. It is not surprising, therefore, that 
World War II has demonstrated the effectiveness of 
assigning a group of scientists to study warfare itself. 

The present volume is a first attempt to describe 
some of the methods which have proved most valu- 
able in the study of warfare, and to indicate possible 
fruitful lines for further development, military and 
nonmilitary. This first chapter outlines the scope and 
methods of the subject. The second chapter discusses 


the relevant portions of the theory of probability, 
which is the field of mathematics most useful for this 
work. The rest of the chapters discuss techniques 
which have been particularly useful, with illustra- 
tions picked from work done in the recent war. One 
aspect of this work, the theory of search for an enemy 
craft, has grown to such considerable proportion that 
it deserves more space than can be given it here. An- 
other volume of this series will be devoted to the 
theory of search and screening (see Division 6, 
Volume 2B). 

11 SCOPE OF OPERATIONS RESEARCH 

The scope of operations research has been suc- 
cinctly illustrated in a letter (dated August 17, 1945) 
from Fleet Admiral E. J. King to Secretary of the 
Navy James V. Forrestal, concerning the work of the 
U. S. Navy Operations Research Group: 

Since April 1942 the Operations Research Group has been 
of service to the Navy as a scientific advisory group to the 
forces afloat and to the Commander in Chief, United States 
Fleet, and Chief of Naval Operations, dealing with naval 
scientific evaluation from the point of view of the operational 
user of naval equipment. This group has been of active 
assistance in: 

(a) The evaluation of new equipment to meet military re- 
quirements. 

(b) The evaluation of specific phases of operations (e.g., gun 
support, AA fire) from studies of action reports. 

(c) The evaluation and analysis of tactical problems to 
measure the operational behavior of new material. 

(d) The development of new tactical doctrine a to meet spe- 
cific requirements (e.g., anti-submarine screens, screens for 
slow moving damaged ships, etc.) 

(e) The technical aspects of strategic planning. 

(f) The liaison for the fleet with the development and re- 
search laboratories, naval and extra-naval. 

This description again emphasizes the important 
point that operations research is not a study of 
equipment, but of operations which involve equip- 
ment and men. 

“Doctrine may be defined as a cumulation of principles, ap- 
plicable to a subject, that have been developed through ex- 
perience, and theory, that represent the best thought of the 
units concerned, and that indicate and guide but do not bind 
in practice. 1 


CONFIDENTIAL 


1 


2 


INTRODUCTION 


1.1.1 Reasons for Increase in Importance 

of Science in Warfare 

There are a number of reasons why the scientific 
study of warfare paid more practical dividends in 
World War II than in previous ones. In the first 
place, recent wars have tended more and more to be 
made up of a series of similar operations rather than a 
set of disconnected battles. The operation of strategic 
bombing is very much the same from mission to mis- 
sion; the target and the weather change, but the 
enemy antiaircraft and fighter defense are roughly 
similar. The operations of submarines in attacking 
enemy shipping are also, in a general sense, repeated 
experiments in tactics. Even landing operations oc- 
curred often enough in World War II, under roughly 
similar conditions, so that one could begin to look for 
similarities as well as differences between these 
actions. With data available on the results of such 
large sequences of roughly similar operations, it be- 
came possible to study the operations in a quantita- 
tive manner. As soon as numbers can be obtained, 
scientific methods can be applied. 

In contrast to World War II, earlier wars consisted 
chiefly in engagements which were strongly condi- 
tioned by the terrain, by the winds, or by other non- 
repetitive agents. Not enough similar engagements 
were fought so that these highly variable aspects 
could be averaged out. Consequently few quantita- 
tive measures could have been obtained, and tactics 
and strategy remained an art, to be learned only by 
long experience in action. 

1.1.2 Increase in Mechanization 

Another reason for the growing usefulness of the 
application of scientific methods to tactics and strat- 
egy lies in the increased mechanization of warfare. 
It has often been said, with disparaging intent, that 
the combination of a man and a machine behaves 
more like a machine than it does like a man. This 
statement is in a sense true, although the full impli- 
cations have not yet been appreciated by most mili- 
tary and governmental administrators. For it means 
that a men-plus-machines operation can be studied 
statistically, experimented with, analyzed, and pre- 
dicted by the use of known scientific techniques just 
as a machine operation can be. The significance of 
these possibilities in the running of wars, of govern- 
ments, and of economic organizations cannot be over- 
emphasized. 


1.1.3 Speed of Technical Advance 

A third reason for the increasing importance of 
operations research in warfare is the increasing tempo 
of obsolescence in military equipment. In World War 
II a number of potentially useful weapons became 
obsolete before the forces at the front had had time to 
learn how to operate them efficiently. When this 
condition holds, it becomes extremely important to 
be able to plan the best employment of each weapon 
ahead of time. When we can no longer have the time 
to learn by lengthy trial and error on the battlefield, 
the advantages of quantitative appraisal and plan- 
ning become more apparent. 

This aspect of the situation is pointed out in the 
following excerpt from the Final Report by Admiral 
E. J. King to the Secretary of the Navy, issued 
December 8, 1945: 

The complexity of modern warfare in both methods and 
means demands exacting analysis of the measures and counter- 
measures introduced at every stage by ourselves and the 
enemy. Scientific research can not only speed the invention 
and production of weapons, but also assist in insuring their 
correct use. The application, by qualified scientists, of the 
scientific method to the improvement of naval operating tech- 
niques and material, has come to be called operations research. 
Scientists engaged in operations research are experts who ad- 
vise that part of the Navy which is using the weapons and 
craft — the fleets themselves. To function effectively they must 
work under the direction of, and have close personal contact 
with, the officers who plan and carry on the operations of war. 

During the war we succeeded in enlisting the services of a 
group of competent scientists to carryout operations research. 
This group was set up as a flexible organization able to re- 
assign personnel quickly when new critical problems arose. 
Fiscal and administrative control of the group was originally 
vested in the Office of Scientific Research and Development. 
The group as a whole was assigned to the Navy for functional 
control, and in the course of time was attached to my Head- 
quarters. 

The initial impulse toward the formation of such a group 
arose in April 1942, during the early days of the anti-submarine 
war. With the cooperation of the Anti-submarine Division of 
the National Defense Research Committee, seven scientists 
were recruited by Columbia University and assigned to the 
Anti-submarine Warfare Unit, Atlantic Fleet. 

During the year 1942 the group was considerably increased 
in size, and in July 1943, at a strength of approximately forty 
members, it was incorporated into the staff of the Tenth Fleet 
as the Anti-Submarine Warfare Operations Research Group. 
Subsequently the administrative responsibility for the group 
was transferred from Columbia University to the Office of Field 
Service, without alteration in relationships with the Navy. In 
October 1944, with the decline of the submarine menace, the 
group was transferred to the Readiness Division of my Head- 
quarters and renamed the Operations Research Group. At the 
close of the war it consisted of seventy-three scientists, drawn 


CONFIDENTIAL 


SCOPE OF OPERATIONS RESEARCH 


3 


from a wide variety of backgrounds. Many of the members 
were attached, as the need arose, to the staffs of fleet and type 
commanders overseas, and at operating bases in war theaters. 
So far as possible they were afforded the opportunity of observ- 
ing combat operations at first hand. 

Operations research, as it developed, fell into two main 
categories: theoretical analysis of tactics, strategy and the 
equipment of war on the one hand; and statistical analysis of 
operations on the other. Each type of naval operation had to 
be analyzed theoretically to determine the maximum poten- 
tialities of the equipment involved, the probable reactions of 
the personnel, and the nature of the tactics which would com- 
bine equipment and personnel in an optimum manner. Action 
reports, giving the actual results obtained in this type of oper- 
ation, were studied in a quantitative manner in order to 
amplify, correct, and correlate closely the theoretical analysis 
with what was actually happening on the field of battle. The 
knowledge resulting from this continued cross-check of theory 
with practice made it possible to work out improvements in 
tactics which sometimes increased the effectiveness of weapons 
by factors of three or five, to detect changes in the enemy’s 
tactics in time to counter them before they became dangerous, 
and to calculate force requirements for future operations. 

The late war, more than any other, involved the interplay of 
new technical measures and opposing countermeasures. For 
example, the German U-boats had to revise their tactics and 
equipment when we began to use radar on our anti-submarine 
aircraft; and we, in turn, had to modify our tactics and radar 
equipment to counter their changes. In this see-saw of tech- 
niques the side which countered quickly, before the opponent 
had time to perfect the new tactics and weapons, had a decided 
advantage. Operations research, bringing scientists in to 
analyze the technical import of the fluctuations between meas- 
ure and countermeasure, made it possible to speed up our 
reaction rate in several critical cases. 

In the normal course of military expediency, 
strenuous efforts are put forth to develop and manu- 
facture new technical equipment and weapons. These 
must be distributed to the operating commands, with 
but limited time in which to develop the best methods 
for their tactical use. As the weapons get into use, 
there begins to flow back an increasing mass of data 
in the form of action reports, performance sheets, and 
intelligence summaries, which are accumulated in 
local commands or at theater headquarters. The sig- 
nificance of the experience embodied in these data can 
only be evaluated through the determined efforts of 
properly trained men with scientific backgrounds, 
clothed with sufficient authority, facilitated by proper 
administrative organization, and freed from other 
responsibilities to concentrate their efforts to this 
end. Such evaluation will indicate the relative merits 
or deficiencies of the new equipment and ordnance, 
and of the tactics incident to their use. In addition to 
indicating means of technical improvements, the 


analyses should also serve as bases for promoting im- 
proved training methods, to make more efficient the 
performance of operating personnel. 

1 * 1.4 Scope of Work 

The nature of research work is admittedly not 
subject to exact definition. Without attempting to 
set a limit to the point where operations research 
merges into developmental work, there are certain 
bounds within which the work of an operations re- 
search group is defined. 

The main function of an operations research group 
is to analyze actual operations, using the data to be 
found in action reports, track charts, dispatches, in- 
telligence summaries from actual interviews of oper- 
ating personnel, etc. The operations research man 
studies weapons and equipment from the “user” 
point of view, both in trials and in field use. He is not 
a statistical clerk; statistical analysis is merely one 
of the tools which he uses in his analytical task. In 
order to extract the best operational use of new weap- 
ons and gear coming into service, it is necessary not 
only to evaluate the prospective performance from a 
priori approaches, but also to analyze observational 
data from experimental and testing establishments 
as well as from actual military operations, no matter 
how complicated. The tactical possibilities and limi- 
tations of any such equipment must be studied in 
relation to the actual facts of military operations. 
Such a performance analysis is essential to a decision 
concerning the requirements for a new weapon or 
device, and to the discovery of the most profitable 
channel in which to devote developmental efforts. 

It must be emphasized again that the operations 
research worker is not a “gadgeteer,” spending most 
of his time devising new equipment or modifying old 
equipment. Such activity is important, and also re- 
quires technically trained men to help the services; 
but it is an activity which should be carried on pri- 
marily by scientists attached to the service com- 
mands or bureaus, rather than by men attached to 
the operational commands, as are the operations re- 
search men. The operations research worker must 
often resist the urge to turn from the refractory prob- 
lems of strategy and tactics, which are primarily his 
job, to the more congenial task of “playing” with a 
piece of new equipment. True, there are times when 
the operations research man is the only technical 
man with an operational command, and equipment 
modifications must be done by him or it does not get 


CONFIDENTIAL 


4 


INTRODUCTION 


done. In this case, of course, the necessary “gadg- 
eteering” is carried through. However, the operations 
research man must keep in mind, in these cases, that 
he is stepping out of his own field, and that such 
activity should not be allowed to keep him long from 
his proper studies of operations. It is possible to call 
in technical men from the service commands to do 
the gadgeteering, but there is no one else to do the 
operations research if he does not do it. 

An important difference between operations re- 
search and other scientific work is the sense of 
urgency involved. In this field a preliminary analysis 
based on incomplete data may often be much more 
valuable than a more thorough study using adequate 
data, simply because the crucial decisions cannot 
wait on the slower study but must be based on the 
preliminary analysis. The worker cannot afford to 
scorn superficial work, for wars do not wait for ex- 
haustive study (although the exhaustive study should 
also be made, to back up the preliminary work) . This 
is an additional reason for divesting the operations 
research worker from extraneous responsibilities and 
duties, so that he can have as much leisure time as 
possible to make as thorough a study as can be made 
before the crucial decision must be made. 

12 METHODS OF OPERATIONS 
RESEARCH 

The methodology of this new application of science 
is, of course, related to the type of data which can be 
obtained for study; in the present case, it usually 
turns out that a limited amount of numerical data 
is ascertainable about phenomena of great com- 
plexity. The problems are therefore somewhat nearer, 
in general, to many problems of biology or of eco- 
nomics rather than to most problems of physics, 
where usually a great deal of numerical data is 
ascertainable about relatively simple phenomena. 
However, operations research, like every science, 
must not copy in detail the technical methods of 
other sciences, but must work out methods of its own, 
suited to its own special material and problems. The 
object here is to assist in the finding of means to im- 
prove the efficiency of operations in progress or 
planned for the future. To do this, past operations 
are studied to determine the facts, theories are elab- 
orated to explain the facts, and finally the facts and 
theories are used to make predictions about future 
operations. This procedure insures that the maximum 
possible use is made of all past experience. 


1.2.1 Statistical Methods 

The most important single mathematical tool of 
operations research is probability and statistical 
theory. The data upon which the research is based 
will come, for the most part, from statistical studies 
of operations. These operations are uncontrolled in 
the scientific sense, and therefore cannot be consid- 
ered as the equivalent of experiments. The data are 
observational, therefore, rather than experimental; 
as is usual in such cases, ingenuous statistical tech- 
niques may lead to serious errors in the results ob- 
tained. 

Statistical analysis is not fruitful unless there are 
available for study a large number of reports on 
operations which are roughly similar in nature. For 
this reason, operations research is at first most suc- 
cessful in those fields of warfare where the individual 
operations are numerous, simple, and roughly simi- 
lar. Bombing operations on a target of a given type 
satisfy these requirements, for a great number of such 
operations are carried out under similar weather con- 
ditions and under somewhat similar conditions of 
enemy opposition. Other examples have been men- 
tioned earlier in this chapter. 

When we come to the larger actions of warfare, 
however, the occasions are few, the complexities of 
the action are great, and the dependence of the out- 
come on the exigencies of the moment are obvious. 
In studying such actions in a statistical manner, the 
progress of operations research will naturally be slow. 
Certain aspects of these larger operations can be 
studied in detail, and the whole picture perhaps can 
eventually be put together. From what has been said, 
however, it can be appreciated that the classical 
naval engagement between surface vessels, or the 
classical land battle, is, in general, difficult for opera- 
tional research to attack early in its development. 
Only after the simpler cases mentioned above have 
been thoroughly understood, can an attempt be made 
to analyze these more complex operations. Strategy 
and tactics in the large will always be an art, though 
operations research may help its practice by provid- 
ing tools of increasing power, just as the study of 
physiology has improved the art of medicine. 

1.2.2 Field Assignments, Collection of Data 

The rapid collection of operational data in wartime 
is immeasurably improved by the assignment of a 
scientifically trained observer as close to the opera- 


CONFIDENTIAL 


METHODS OF OPERATIONS RESEARCH 


5 


tions as is feasible. In earlier centuries this was quite 
possible because of the small scope of most opera- 
tions. In the World War I however, it became 
exceedingly difficult, primarily because the scope of 
the operations was so large that, if the scientist got 
close enough to see the details of operations, he in- 
evitably became a participant rather than an ob- 
server. The problem has become somewhat more 
simplified in World War II, due primarily to the 
introduction of the airplane. In all combat involving 
aircraft, the technical observer can be placed at a 
forward air base and get his reports at first hand from 
the participants immediately after they have re- 
turned from the operations. It has been found by 
experience that important facts concerning the opera- 
tions can often only be determined by having a tech- 
nically trained observer question the operational per- 
sonnel at first hand. 

Another important function of the men in the field 
is to see that the usual action reports contain as much 
useful data as possible. Because these men know the 
kind of data that are most amenable to analysis, they 
can try to see that the reports are as complete as 
possible and are as “painless” as possible for the per- 
son making them out, and that the reports get sent 
as quickly as possible to the headquarters group for 
analysis. They can also detect at first hand what 
kinds of intelligence material are likely to be unre- 
liable, due to local factors which are not always ap- 
preciated at headquarters. 

The observers’ reports, together with the usual 
operational reports, must then be sent in to a central 
group which analyzes the results from all theaters and 
compares them for differences and similarities. The 
importance of the close interrelation between the field 
observers and the central group is obvious. In prac- 
tice it has been arranged that members of the central 
group spend a certain part of their time in the field, to 
return later to the central group with increased in- 
sight into the operations they are studying. 

1-2.3 Limitations of Operational Data 

Statistical analysis is part of the observational 
aspect of operational research. The operations are not 
controlled in the scientific sense, and insight into the 
reasons for their success or failure can only be ob- 
tained by studying large numbers of similar opera- 
tions, so as to find out by statistical methods the 
effects of the variation of one or more components of 
the operation. This imposes certain limitations on the 


usefulness of the results of the statistical analysis, for 
the range of variation of the various components in 
the operation will, by the nature of things, be rather 
limited. Once successful tactics have been devised, it 
becomes less and less likely that the opponents in the 
individual operation will deviate widely from the ac- 
customed mean. Consequently the operational data 
can only be utilized (by calculus of variations meth- 
ods) to find whether small changes in components 
will improve or diminish the results. 

The results of such variational studies are quite 
useful, and the applications sometimes quite strik- 
ing, since often the enemy’s reaction to a quantitative 
change in our operation is a qualitative change in his 
counteraction. However, a study of small variations 
is not usually sufficient. In many cases what is inter- 
esting is not a small change in the tactics used but a 
completely different combination of actions, a “mu- 
tation” of the operation, as it were. These new tactics 
may not be predictable from the old operations by 
variational calculation, for the extrapolations may be 
too large for first order terms. Here a purely mathe- 
matical analysis of the whole operation, or of parts of 
it, may supply the necessary knowledge; or it may be 
possible that a series of discussions with the operating 
personnel may bring the necessary insight. 

1-2.4 Limitations of “Expert Opinion” 

It should be mentioned, however, that the opinions 
of a few dozen persons who have had operational ex- 
perience provide an extremely shaky foundation for 
any operations research. It is unfortunately true, 
though not often realized, that people seldom esti- 
mate random events correctly; they always tend to 
remember the “exciting one” and forget others, and 
as a result their opinions are nearly always uncon- 
sciously biased. Their actions in an operation or an 
operational experiment are important and worth re- 
cording, however. The need for unbiased, impersonal 
facts, not opinions, must always be borne in mind; 
military personnel (and indeed most people without 
rigorous scientific training) tend to take the opposite 
opinion of the relative validity of opinion versus 
facts. One often hears the question, “Why do you 
need detailed action reports (or why should you wit- 
ness this operation) when so-and-so can tell you all 
about it?” If science has learned one thing in the past 
three centuries, it is that such a point of view must 
be avoided if valid scientific results are to be achieved. 

The statistical analysis of past operations is a 


CONFIDENTIAL 


6 


INTRODUCTION 


vitally important part of operations research, but it 
has its limitations, and it must be supplemented by 
other methods of scientific attack. 

1-2.5 Operational Experiments 

In a few cases experimental methods can be used. 
A tactical exercise can be laid out so that quantitative 
measurements of the behavior of the forces engaged 
may be obtained. Such controlled experiments are 
often difficult to arrange, so that they are really 
measured experiments rather than training exercises; 
for this reason not many of them have been carried 
out. Many more could and should be carried out; per- 
haps the most useful activity of operations research 
in peacetime will be the organization and study of 
such tactical experiments. 

Although operational experiments usually deal 
with simplified components of an operation, this is 
not an argument that such experiments have little 
value, though it is an argument that such experiments 
must be designed very carefully in order to produce 
useful quantitative results. As a matter of fact, oper- 
ational experiments have proven to be a most valu- 
able source of quantitative data concerning opera- 
tions, and it is highly important that such experi- 
ments be continued in greater number in the future. 
There is great need, in particular, for further study in 
the techniques of planning these tactical experiments 
and in methods of measuring the results. These 
matters will be discussed in detail in Chapter 7. 

1.2.6 Analytical Methods 

Finally, operations research must also use purely 
theoretical methods in its development. In fact, if it 
is to progress as any other branch of science, its aim 
must be to transform as rapidly as possible the 
empirical data which it collects into generalized 
theories which can then be manipulated by mathe- 
matical methods to obtain other results. This aim is 
just as true of the biological sciences (of which opera- 
tional research is a minor member) as it is of the 
physical sciences, although the progress is more 
difficult. The work of J. B. S. Haldane, 2 R. A. Fisher, 3 
and others is a good example of the power of theo- 
retical methods in genetics. A certain amount of 
analogous theoretical work has been done in opera- 
tions research on the effect of various strategic distri- 
butions of forces. This will be reviewed in Chapter 4. 


An important element which enters into the theo- 
retical treatment of tactics and strategy is the one of 
competition between the opposing forces. The system 
as a whole cannot be considered as a purely mechan- 
ical one with single responses to specific situations. 
The recent work of Von Neumann and Morgenstern 4 
indicates that even this element of competition can 
be handled mathematically in an adequate manner. 
Some aspects of their work will be discussed in 
Chapter 5. 

The fact that at present purely theoretical analyses 
of strategy and tactics confine themselves to ex- 
tremely simplified components of operations must 
not blind us to the importance of such studies and to 
their eventual practical utility. The theoretical 
aspects of every science must start with the study of 
absurdly simplified special cases. When these simple 
cases are fully understood, and are then compared 
with the actualities, further complexities can be intro- 
duced, and cases of practical importance can eventu- 
ally be studied. As has been cogently said, 4 “The 
mechanical problem of the free fall of bodies is a very 
trivial physical phenomenon, but it was the study of 
this exceedingly simple fact, and its comparison with 
the astronomical material, which brought forth me- 
chanics.” 

It can thus be seen that operations research at 
present is but the beginning of the science of warfare. 
The start has been made, however; the statistical 
methods are at least known; the techniques of tac- 
tical experimentation have been partly developed. 
Further mathematical techniques may need to be 
developed, but even here the first steps can be 
sketched out. The practical usefulness of the results 
so far obtained emphasizes the importance of further 
development. 

13 APPLICATIONS TO GOVERNMENT 
AND INDUSTRIAL PROBLEMS 

Operations research, as it has been known hitherto, 
is the application of the scientific technique to the 
study of combinations of men and equipment in war- 
fare. As such, it is an application of science to the 
business of destruction and the defense against enemy 
destructive power. While it is necessary, in the 
present absence of world government, such applica- 
tions are basically at variance with the fundamental 
creative urge of most scientists. There is therefore 
somewhat of a tinge of self- justification in raising the 


CONFIDENTIAL 


APPLICATIONS TO GOVERNMENT AND INDUSTRIAL PROBLEMS 


7 


question here as to whether the techniques of opera- 
tions research could be used in the study of peacetime 
operations. 

There are other reasons for raising the question of 
peacetime applications, however, besides the natural 
wish to apply science to peaceful activities. Scientists 
engaged in operations research during the war have 
nearly all of them been struck by two simple but sur- 
prising facts: 

1. Large bodies of men and equipment carrying out 
complex operations behave in an astonishingly regu- 
lar manner, so that one can predict the outcome of 
such operations to a degree not foreseen by most 
natural scientists. 

2. The lack of understanding of the scientific ap- 
proach to a problem, and the lack of awareness that 
scientific methods can help solve operational prob- 
lems, on the part of most governmental, military, and 
industrial administrators, is likewise astonishing (as 
well as depressing) to natural scientists. 

The first of these two observations indicates 
strongly that the point of view and methods of opera- 
tions research can be of great value in fields other 
than warfare. The second point indicates certain fun- 
damental differences of attitude between scientists 
and administrators that require careful organiza- 
tional planning and “indoctrination” of both scien- 
tists and administrators, if operations research is to 
be of use in a given field. These problems were met 
and solved for the military field during the past war: 
they are discussed later in this section so they can be 
recognized and solved in other fields. 

1 . 3.1 The Constancy of Operational Constants 

The uniformity of behavior of equipment and men 
is not usually apparent until a number of similar 
operations have been compared, for this uniformity is 
usually of a statistical nature. For instance, the frac- 
tion of U-boats, sighted by British aircraft in day- 
light, which are attacked while still visible, was found 
to be about 40 per cent in the summer of 1941. This 
fraction remained nearly constant, fluctuating be- 
tween 35 and 50 per cent, for nearly two years. When 
the United States entered the war, similar calcula- 
tions on this side of the Atlantic showed that this 
same fraction was between 40 and 45 per cent for all 
of 1942, even though the plane type and the training 
methods differed from the British. Other operational 
quantities which have remained constant for long 
periods of time are discussed in Chapter 3. 


This constancy (and therefore predictability) of 
operations involving men and equipment is at least 
in part due to the fact that the presence of equip- 
ment forces the men to act in a mechanical manner. 
This may be deplored by poets, but it should be the 
basis of hope for social scientists, economists, and 
systems engineers. It means that, within certain 
limitations, the behavior of men and machines can be 
experimented with, measured, computed, and pre- 
dicted. Viewed from this more general point of view, 
the technique of operations research can certainly be 
applied to the study of peacetime operations. 

Such studies will draw upon both the physical and 
biological sciences. The laws governing the sighting 
of a U-boat by an aircraft observer depend upon the 
visual contrast between the normal ocean surface and 
the wake of the submarine, the optical transmission 
properties of the atmosphere, the methods by which 
a person’s eye scans a large area, the sensitivity of 
the eye retina to contrast, and on the altitude and 
speed of the plane. Some of these quantities are 
physical, some physiological; they all are measurable. 
The sighting of the incoming aircraft by a lookout on 
the U-boat depends on similar quantities. The frac- 
tion of U-boats which have not the time to submerge 
before the aircraft attacks them depends on the re- 
lationship between these two sets of quantities. 

From this point of view, the statement that 40 per 
cent of the submarines are attacked on the surface 
corresponds to an interrelation between physical and 
physiological optics and the physical capabilities of 
submarines and aircraft. This point of view is not a 
new one. What has not been usually appreciated is 
that these overall quantities often remain remarkably 
constant, in spite of wide variations of extraneous 
circumstances, as long as the equipment involved re- 
mains approximately the same. As soon as the Ger- 
mans made a radical change in their submarine de- 
sign, the fractign mentioned above changed markedly 
in value. 

Similar overall quantities can probably be deter- 
mined for peacetime operations. Preliminary studies 
of automobile traffic indicate that this is amenable to 
such analysis. The combined behavior of telephone 
subscribers, telephones, telephone operators, and 
switchboards has long been studied by mathemati- 
cians at the Bell Telephone Laboratories. 5 The be- 
havior of customers in a restaurant or of workers in a 
factory, the reaction of readers to an advertisement 
or of the population to a subway system should all 
yield operational quantities which can be used for 


CONFIDENTIAL 


8 


INTRODUCTION 


future prediction. In some of these fields efficiency en- 
gineers and others have already made a beginning, 
though the interrelations between the fields have not 
been clarified as yet. 

1.3.2 Analytical Study of Operational 
Constants 

From the foregoing, one might think that opera- 
tions research was but a military application of what 
is sometimes known as “efficiency engineering.” If 
operations research had stopped at getting overall 
constants, such as the fraction of submarines at- 
tacked on the surface, this statement might be valid. 
In most cases, however, it was found particularly 
fruitful to continue the study into the details of the 
operation. In the aircraft-versus-U-boat case, for in- 
stance, it proved advantageous to set up physiological 
experiments to determine the behavior of the human 
eye when “scanning the horizon.” Curiously enough, 
such experiments had not been done in detail before, 
and the work resulted in important and valuable new 
knowledge concerning the use of the eye. This new 
knowledge could be put back into the original prob- 
lem to gain a much broader picture of the mechanism 
of sighting U-boats from aircraft, and vice versa. 
These facts, together with optical measurements of 
the atmosphere and of properties of the ocean sur- 
face, enable one to put together a fairly complete 
theory of aircraft search by means of which one can 
predict what would happen in case the equipment 
were changed. 

The scientific studies, therefore, went in two steps: 
first there was the recognition of the constancy of 
certain quantities typical of the operation, by means 
of which one could predict the outcome of future 
operations as long as the equipment was unchanged. 
Finally, after subsidiary laboratory measurements 
were made, a fairly detailed theory of the whole 
operation is obtained, from which one can predict the 
outcome of future operations even if the equipment is 
changed. One is now in a position to determine what 
modifications of equipment or procedure are neces- 
sary to obtain the best results from the operation 
under study. Thus, by a judicious combination of a 
statistical analysis of the overall results of an opera- 
tion with purely physical and physiological labora- 
tory experiments, one arrives at a truly scientific 
insight into the details of the operation. When this 
stage has been reached, it is possible to “design” the 
operation to give the results desired. 


1-3.3 Measures of Value 

At this stage in the investigation the worker must 
broaden his field of view and investigate the measures 
of value which must be applied to the operation. As 
soon as a detailed theory enables one to predict the 
changes in overall results arising from changes in 
equipment or procedure, one must ask which result is 
better than the other and by how much is it better. In 
the case of operations of war, the measure is often 
easy to find : the operation is best which destroys the 
enemy force most rapidly or which destroys most of 
his productive capacity, etc. Measures of value for 
peacetime operations are sometimes as easy to deter- 
mine, but in many other cases they are not. For 
instance, should an automobile traffic system be de- 
signed to transport people from the suburbs to the 
center of town as rapidly as possible, or to permit the 
delivery of the greatest amount of goods by truck, or 
to produce the fewest deaths? Should a housing de- 
velopment be designed to be cheapest for the buyer, 
cheapest for the community to service, or most 
profitable for the builder? 

Questions of measures of value are rather unfa- 
miliar to the physical scientist, but it is essential that 
he study them when engaged in operations research. 
Experience in the war has shown that the scientist 
himself must usually discover the proper measure of 
value just as he must often discover the nature of the 
problem itself. This brings us back to the second 
point, mentioned earlier, that administrative officials, 
military or nonmilitary, seldom realize that their 
operational problems can be dealt with in a scientific 
and quantitative manner. This is not surprising, since 
it was not realized by scientists themselves how many 
such problems were amenable to study. But it does 
mean that a considerable amount of initial effort 
must often be spent in persuading the higher com- 
mand that some of their problems can be solved by 
quantitative means and in acquainting them with the 
methods whereby these solutions are obtained. 

1.3.4 Importance of Mutual Understanding 
between Administrator and Scientist 

The reaching of a working understanding on “terms 
of reference” between the operations research worker 
and the administrative head to whom he is assigned 
is one of the most important organizational problems 
encountered in entering a new field of operations re- 
search. Scientist and administrator perform different 


CONFIDENTIAL 


APPLICATIONS TO GOVERNMENT AND INDUSTRIAL PROBLEMS 


9 


functions and often must take opposite points of 
view. The scientist must always be skeptical, and is 
often impatient at arbitrary decisions; the adminis- 
trator must eventually make arbitrary decisions, and 
is often impatient at skepticism. It takes a great deal 
of understanding and mutual trust for the two to 
work closely enough together to realize to the fullest 
the immense potentialities of the partnership. 

These psychological difficulties are pointed out here 
so they can be foreseen and allowed for in the future. 
During World War II they often caused confusion 
and inefficiency because they were not appreciated. 
Fundamentally, the problem is to convince the ad- 
ministrator that the scientist can help him make his 
decisions more effectively and wisely, and to con- 
vince the scientist that the administrator is still the 
one to make the basic decisions. 

The first reaction of the administrator to operations 
research is usually that the scientists are welcome, 
but that there seems to be no important problem 
which is suitable for them to attack. Next comes 
a reaction of suspicion and impatience, when the 
usual scientific procedure of scrutinizing critically all 
assumptions is commenced. Considerable tact must 
be employed to persuade the administrator that 
measures of value and estimates of results are not 
called in question simply from a desire to criticize. 

At this stage in the proceedings great care must be 
exercised to keep the initial doubts and questionings 
(necessary for the scientific analysis) from spreading 
to other parts of the organization. Once a few success- 
ful solutions have been obtained and the command 
realizes that all of this critical questioning does pro- 
duce results, the worst of the opposition is over. In 
time the commander comes to recognize the sort of 
problems which operations research can handle and 
comes to refer these problems to the group without 
prodding from the group itself (at least until the 
normal rotation brings a new set of officers who must 
be indoctrinated anew) . 

Occasionally there is some suspicion that the oper- 
ations research worker wishes to take over the com- 
mand function of the officer. This may come up if the 
findings of the operations research worker are con- 
siderably at variance with the preconceived opinion 
of the officer. This suspicion can only be overcome if 
both the worker and the officer realize that the results 
of operations research are only a part of the material 
from which final decision must be made. In any ad- 
ministrative decision there enter a great number of 
considerations which cannot be put into quantitative 


form (or at least cannot yet be put into this form). 
Knowledge of these qualitative aspects, and ability 
to handle them, is the proper function of the ad- 
ministrator, and is not the prerogative of operations 
research. The operations research worker, unless he 
is to operate in a dual role of scientist and adminis- 
trator, must work out those aspects of the problem 
which are amenable to quantitative analysis and 
report his findings to the administrator. The adminis- 
trator must then combine these findings with the 
qualitative aspects mentioned above, to form a basis 
for the final decision. This decision must be made by 
the executive officer. If his decision runs counter to 
the scientific findings at times, the scientist must not 
consider that this is necessarily a repudiation of his 
work. 

1.3.5 Possible Peacetime Applications 

Very much the same sort of initial opposition can 
be expected from governmental and industrial ad- 
ministrators. Once this is overcome, however, there 
is no reason why operations research should not be as 
fruitful in aiding in the solution of these problems as 
it was in helping solve military problems. Just as with 
problems of war, of course, some operations will be 
much more fruitful of results than others. Traffic 
problems, for instance, are highly amenable, for data 
are easy to obtain, and changes in conditions (if not 
too drastic) can be produced to study the effects. 

On the other hand, the design of city housing and 
municipal facilities requires data which are difficult 
to obtain, the solution is strongly dependent on ter- 
rain and other individual circumstances, and opera- 
tional experiments are difficult if not impossible. The 
field of housing and of city planning is an extremely 
important one, however; and operations research in 
this field should be started as soon as an adequate 
administrative authority is set up to whom the scien- 
tist could report, and which could ensure that the 
research is more than idle academic exercise. Opera- 
tions research in telephone operation is not difficult 
because the whole system is under a more or less uni- 
fied control. (In fact, operations research in this field 
has been going on for a number of years under the 
name of systems engineering.) Operations research in 
house-heating, however, might well be fruitless, be- 
cause the fragmented nature of the industry makes 
the gathering of data difficult and makes any action 
on proposed solutions well-nigh impossible. (A ques- 
tion which might be important, but which would be 


CONFIDENTIAL 


10 


INTRODUCTION 


difficult to answer, would be : if another war is likely 
to occur in ten years, should this country encourage 
coal heating or oil heating or electric heating in the 
homes in northeastern United States?) Operations 
research on traffic might well result in suggestions for 
change in design of automobiles, but the competitive 
nature of this industry would make it extremely diffi- 
cult for the suggestions to be put into practice. 

All of these comments serve to emphasize the ob- 
vious fact that operations research is fruitful only 
when it studies actual operations , and that a partner- 
ship between administrator and scientist, which is 
fundamental in the process, requires an administrator 


with authority for the scientist to work with. Opera- 
tions research done separately from an administrator 
in charge of operations becomes an empty exercise. 
To be valuable it must be toughened by the repeated 
impact of hard operational facts and pressing day-by- 
day demands, and its scale of values must be repeat- 
edly tested in the acid of use. Otherwise it may be 
philosophy, but it is hardly science. 

It is hoped that operations research in peacetime 
fields will be carried on in the next few years to inves- 
tigate how real are the difficulties mentioned above, 
and to demonstrate (perhaps) that this aspect of 
science can be as valuable in peace as in war. 


CONFIDENTIAL 


Chapter 2 

PROBABILITY 


T he theory of probability is the branch of 
mathematics which is most useful in operations 
research. Nearly all results of operations of war in- 
volve elements of chance, usually to a large extent, so 
that only when the results of a number of similar 
operations are examined does any regularity evidence 
itself. It is nearly as important to know the degree by 
which individual operations may differ from some 
expected average, as it is to know how the average 
depends on the variables involved. In analyzing oper- 
ational data, which are often meager and fragmen- 
tary, it is necessary to be able to estimate how likely 
it is that the next operations will display character- 
istics similar to those analyzed. Probability enters 
into many analytical problems as well as all the sta- 
tistical problems. 

The present chapter will sketch those parts of the 
theory of probability which are of greatest use in 
operations research, and will illustrate the theory 
with a few examples. Section 5.1 will deal in detail 
with specific methods of handling statistical prob- 
lems, and Chapters 6 and 7 will deal with some of the 
applications of probability theory to analytical prob- 
lems. For further details of the theory, the reader is 
referred to texts on probability theory. 5 

2.1 FUNDAMENTAL CONCEPTS 

In many situations the system of causes which 
lead to particular results is so complex that it is im- 
possible, or at least impracticable, to predict exactly 
which of a number of possible results will arise from a 
given cause. If a penny is tossed, it is possible in 
principle to analyze the forces acting on the penny 
and the motions they produce, and so to predict 
whether the penny will come to rest with heads or 
tails showing; however, no one has ever taken the 
effort to carry out the analysis. When a gun is fired 
at a target, it should again be possible to predict 
exactly where the shell will hit, but the prediction 
would involve a knowledge of the characteristics of 
the gun, shell, propellant, and atmosphere far more 
exact than has yet been obtained. 

With a perfect penny, tossed at random, there is 
no more reason to expect heads than tails to appear. 
We say then that heads and tails are equally likely to 


appear. In throwing a symmetrical die the numbers 
1, 2, 3, 4, 5, and 6 are equally likely. This notion of 
equal likelihood is basic to the theory of probability. 
It does not seem to be possible to give it an exact 
definition, but we accept it as a self-evident intuitive 
concept. At times (as with a coin or die) we reach the 
conclusion that two results are equally likely from 
considerations of symmetry. In other cases the con- 
clusion is made on the basis of past experience. Thus, 
for example, if a gun is fired a great number of times, 
and right and left deflections appear an equal num- 
ber of times, we reach the conclusion that right and 
left deflections are equally likely. 

From the notion of equal likelihood we can derive 
the idea of randomness. Suppose that we have a 
chance method by which a point is chosen on a line 
of finite length. If the method is such that the point is 
equally likely to fall in any of a number of parts of 
the line of equal length, we say that the point is 
chosen at random. For example, if a perfectly bal- 
anced wheel is spun hard and allowed to come to rest 
under the action of a small amount of friction, the 
point of the circumference which comes to rest under 
a stationary index pointer is a random point of the 
circumference. Or a random point may be chosen by 
drawing a series of numbers from a hat containing 
slips of paper with the digits 0, 1, 2, • • -9 (replacing 
the slip after each drawing), and writing the result as 
a fraction in decimal notation. This fraction is then 
the coordinate of a random point on a line of unit 
length. Examples of random sequences of numbers 
are given in Tables I and II on pages 153 to 155. 

We may also speak of points chosen at random in 
spaces of more than one dimension. Thus, for exam- 
ple, we may say that a point is chosen at random in a 
given area if, given two parts of the area of equal 
size, the chosen point is equally likely to be in either 
one of them. 

2.1.1 Probability 

If we now consider a situation in which any one of 
a number of results may occur (but not necessarily 
with equal likelihood), we may compare the likeli- 
hoods of these results with the likelihood that a point 
chosen at random on a line falls within a given inter- 


CONFIDENTIAL 


11 


12 


PROBABILITY 


val on that line. In fact, the line may be divided into a 
set of intervals in such a way that each interval cor- 
responds to one of the possible results, and so that the 
likelihood of each result and the likelihood of a 
random point of the line falling in the corresponding 
interval are equal. 

Thus, in the case of the tossed penny, we may com- 
pare the chances of heads and tails with the chance 
that a point chosen at random on a line falls in the 
right or left half of the line. The situation is shown in 

HEADS j TAILS 

I I 1 

0 7 

F 

Figure 1. Comparison of randomly chosen points on a 
line with throws of a coin. 


I I I I I I I 

|1|2|3|4|5|6| 

I 1 1 1 1 1 1 

o JL _2_ _3_ A _5 1 

6 6 6 6 6 

Figure 2. Comparison of randomly chosen points on a 
line with throws of a die. Each y 6 portion of the line cor- 
responds to a face of the die. 

Figure 1. For the rolls of a die, the intervals for the 
possible results may be chosen as in Figure 2. 

When the process has been carried out, the length 
of the interval corresponding to each result, meas- 
ured in terms of the total length of the line as a unit, 
is defined as the 'probability of that result. Thus the 
probability of throwing heads with a coin is and 
the probability of rolling a 3 with a die is %. 

Several theorems concerning probabilities are ob- 
vious from this definition: the probability that one or 
another of a set of possible results will be obtained is 
the sum of the probabilities of the individual results; 
the sum of the probabilities of all the results is unity; 
if p is the probability of any result, the probability 
that the result does not happen is 1 — p, and so on. 

2* 1 * 2 Distribution Functions 

This same definition can be applied when the pos- 
sible results consist of values of a continuous variable. 
Consider the following example. A long rod is pivoted 
at its center and spun. We wish to know the proba- 
bility that when it comes to rest, the rod (or its 
extension) will intersect a given line within any given 
interval (Figure 3A). Let XY be the line, and let AB 


be the rod, pivoted at the point 0, a perpendicular 
distance a from XY. Let x be the distance of the 
point of intersection from the foot of the perpendicu- 
lar from 0 to XY. If 6 is the angle made between the 
rod and a line parallel to XY, then the effect of the 



B 

X 

-oo -2a -a 0 a 2a oo 

[““M 1 

o 1/4 1/2 3/4 1 

F 

Figure 3. A. Rod AB is spun about pivot 0 and comes 

to rest at angle 6, intersecting line XY at point x. B. 

Shows relation between x and F = 1 — 6/ir. 

spinning is to choose a value of 6 at random between 
0 and 7 r. The value of x is then determined by 

x = a cot 0 . (1) 

Since 6 has a random value between 0 and 7 r, then 
F = (ir — d)/ir has a a random value between 0 and 1, 
and 

x = —a cot (7 rF) . (2) 

We may now represent the situation by a diagram of 
the type of Figures 1 and 2 if we take a line of unit 
length and mark it with a uniform scale for the var- 
iable F, and another scale for the corresponding 
values of x. This is shown in Figure 3B. The proba- 
bility that x lies between any two values and x 2 is 
equal to the length of the corresponding interval on 
this scale. Arithmetically this is equal to F 2 — F h so 
that this probability is 

P = F 2 — Fi = — - cot- 1 (-) + - cot- 1 (-) . (3) 
7r \a / 7r \a / 

a The expression (x — 0) /x is chosen instead of the more 
obvious 0/x in order to make x an increasing, rather than a 
decreasing, function of F. 


CONFIDENTIAL 


FUNDAMENTAL CONCEPTS 


13 


We see from the previous problem that we have 
two basic types of variables. This will also hold true 
in the general case. 

The fundamental variable, from the theoretical 
point of view, we will call the random variable £, 
which will have any value (within its allowed range) 
with equal probability. The mechanics of the problem 
must be analyzed sufficiently to say that a random 
trial corresponds to a random choice of £. 

The second type of variable, the stochastic variable 
x will be dependent upon the random variable, that 
is, a random choice of £ will define some value of x. 
The stochastic variable is the quantity we measure 
experimentally. We may write x as some function of 
£ such that the proper relationship holds for all 
values of £ and x. 

For convenience in analyzing the problem, we 
make a choice of origin and scale for the random 
variable such that the values of £ will occur between 
zero and unity. We may do this by suitably combin- 
ing the random variable with the (constant) values 
it takes at the ends of its allowed range. When this is 
done, the values found for £ in the course of many 
trials will be distributed more or less uniformly over 
the interval (0, 1). (In the limit, as the number of 
trials goes to infinity, all possible values of £ from 
zero to unity will occur.) 

If £, which is now defined from zero to unity, is 
represented as a function of x, we may write 

£ = F(z); 0^£^1. (4) 

This function, F(x), is such that the process of choos- 
ing a value of x is the same as choosing a value of £ 
at random in the interval (0, 1). F(x) is then called the 
distribution junction of the variable x. The probability 
that x lies between x\ and x 2 is F(x 2 ) — F(xi ) . 

For an infinitesimal interval dx, located at x, the 
probability that the stochastic variable lies in this 
interval is F(x + dx) — F(x) = (dF/dx)dx (by a 
Taylor’s series expansion). The function f(x) = dF/dx 
is known as the probability density at x. The tradi- 
tional treatment of probabilities in a continuum takes 
the probability density as fundamental. In dealing 
with statistical data, however, working with/(#) in- 
volves the difficulties and inaccuracies inherent in 
numerical differentiation. This can largely be avoided 
by using F(x) instead of f(x). [In addition, if F(x) is 
discontinuous, the probability density has no simple 
meaning.] 

We see how this applies to the previous problem. 
There, 6 is the random variable, with its allowed 


values going from 0 to i r, and x is the stochastic 
variable, where x = a cot 0. The scale and origin of 0 
are then redefined, so that we form the new random 
variable with a range from zero to unity, that is, 




Figure 4. Probability of sighting an object at relative 
bearing a. The distribution function is F, and / is the 
probability density. 


( 7 r — 0)/-7T. This new random variable, when taken as 
a function of the stochastic variable x, is then the 
distribution function for x : 



The probability density is therefore 



Another example, corresponding to a more imme- 
diately useful problem, comes from the theory of 
search (see Division 6, Volume 2B for further de- 
tails). Suppose a search vessel, at 0 in Figure 4, is 


CONFIDENTIAL 


14 


PROBABILITY 


moving with constant velocity in the direction indi- 
cated by the arrow. The object searched for (life raft, 
enemy vessel, etc.) is likely to be anywhere on the 
ocean, and is assumed at rest for simplicity. We make 
the simplifying assumption (which is not a bad one 
for some cases) that if the object comes within a 
radius R of the vessel it will be discovered. The ques- 
tion to be answered here is the probability that the 
object, if it is discovered, comes into view at a rela- 
tive bearing a. 

Relative to the search vessel, the ocean is moving 
along the parallel paths shown in the figure. The 
object will also move along one of these relative 
paths, say the one coming a nearest distance l from 
the search vessel. It is not difficult to see that, if the 
object is placed at random, and if it is to be dis- 
covered, the value of l will occur at random between 
the limits —R and + R. 

We see, therefore, that since all values of l (be- 
tween — R and -fi?) have equal probability, l is the 
random variable. The angle a, which is to be meas- 
ured experimentally, is the stochastic variable, and 
is related to the random variable by l = R sin a. We 
now redefine the random variable so that it takes on 
values between zero and unity, that is, 

£ = ^ ^ l when —R < l < R . 

2 R’ ~ ~ 

We may therefore write for the distribution function 
„ , x R + R sin a 


= \ (1 + sin a) . 

The probability that the object will be sighted be- 
tween the bearings c*i and a 2 (— tt/ 2 < a h a 2 < ir/2) is 

F(a 2 ) — F(a i) = \ (sin a 2 — sin c*i). 

In particular, the probability that the object will be 
sighted between the bearings a and a + da (i.e., will 
be sighted “at the bearing a ” in the element da) is 

f(a)da = \ cos ada. 

The quantity f(a) = ^ cos a is the probability 
density. Both F and / are plotted in Figure 4. 

We see that, as long as our assumptions hold (effi- 
ciency of lookouts equal in all directions, all objects 


sighted at range R), then the object is more likely to 
be sighted in the forward quarter than on either 
beam, since /(a) is largest in this region. Moreover, a 
restriction of the lookouts to searching over the for- 
ward quarter will only reduce the probability of 
sighting by approximately 30 per cent. (In the exam- 
ple considered here, this might be the wrong restric- 
tion to make, for two lookouts facing in opposite di- 
rections and looking out on either beam will eventu- 
ally sight all the targets which all-round-looking 
lookouts could discover. Why? Which restriction is 
best must be decided on other grounds.) 

The distribution function may be applied to dis- 
crete as well as to continuous stochastic variables. 
The rolling of a die, for example, may be thought of 
in terms of a variable x, the number appearing on the 
die, and a distribution function F, related by the 
equations : 


X 

= 1 

0 

< 

F 

< 

1/6 

X 

= 2 

1/6 


F 

< 

2/6 

X 

= 3 

2/6 

< 

F 

< 

3/6 

X 

= 4 

3/6 


F 

< 

4/6 

X 

= 5 

4/6 

< 

F 

< 

5/6 

X 

= 6 

5/6 

<; 

F 


1 


It should be noted that £ is a single- valued function 
of F (although discontinuous), but F is not a single- 
valued function of x. To try to express the proba- 
bility density / = dF/dx in such cases involves 
mathematical difficulties which cannot be discussed 
here. 

In each problem dealt with in the theory of proba- 
bility we are dealing with one or more trials. A gun 
is shot, or a depth charge is tested, or a fighter plane 
encounters an enemy, or a search plane tries to find 
an enemy vessel. In each case we are interested in the 
outcome of the trial or trials, which usually takes the 
form of a numerical result. The range of the shell shot 
from the gun may be the interesting quantity, or the 
depth at which the depth charge exploded, or the 
length of time required to find the enemy vessel. 
Sometimes the answer can be a discrete one; we may 
be interested only in whether the fighter plane was 
shot down or whether it shot down the enemy, or 
whether neither was shot down. This numerical re- 
sult, which may differ from trial to trial, is what is 
called the stochastic variable, x.We are usually inter- 
ested in determining the probability of occurrence of 
different values of this variable for different trials, 
or else we are interested in determining its average 
value for a large number of trials. 


CONFIDENTIAL 


FUNDAMENTAL CONCEPTS 


15 


In a great number of cases these probabilities and 
average values can only be determined experimen- 
tally by making a large number of trials. In some 
other cases, such as the ones considered previously in 
this chapter, it is possible to analyze the situation 
completely and to work out mathematically the ex- 
pected behavior of the stochastic variable at future 
trials. It is possible to make this analysis in a much 
larger number of cases than might be expected; and 
in a great many more cases it is possible to make an 
approximate analysis of the situation which will be 
satisfactory for most requirements. 

2.1.3 Distribution Functions in Several 
Variables 

In more complicated situations, the result of a 
chance process requires more than a single variable 
for its expression. Such cases can be handled in a way 


61 

62 

63 

64 

65 

66 

51 

52 

53 

54 

55 

56 

41 

42 

43 

44 

45 

46 

31 

32 

33 

34 

35 

36 

21 

22 

23 

24 

25 

26 

1 1 

1 2 

1 3 

1 4 

1 5 

1 6 


0 O.fe 1.0 



Figure 5. Representation of probability distribution in 
two variables. Two dice. 

entirely similar to those of the previous section, but 
the distribution function, instead of corresponding to 
points chosen at random on a line, now corresponds 
to points chosen at random in an area, a solid, or a 
figure of a higher number of dimensions. 

As an illustration, consider the throws of a pair of 
dice. The result of each throw can be thought of as 
determined by a random variable, F\ for the first die, 
and F 2 for the second, in the way described in the 
preceding section. For the two together we may 
combine the choosing of Fi and F 2 into the process of 
choosing a point at random in a square, in which Fi 


and F 2 are the two coordinates of the point. Figure 5 
shows such a square. It divides into 36 small squares, 
each corresponding to a single result of the throw. 
Since each of these has an area equal to ^ (the 
area of the large square being unity), the probability 
of any one throw is . It is also easy to see the 
probability of obtaining any given total. There are 
just six squares in which the total is 7, so the proba- 
bility of throwing 7 is or i. 



A 



V (fan) 


Figure 6. Needle on ruled paper (Buffon’s problem). 
Plot of distribution functions versus numbers of lines 
crossed. 

As a second illustration, we may consider the fol- 
lowing problem (Buffon’s needle problem). A sheet 
of paper is ruled with parallel lines a distance a apart. 
A needle of length l is thrown on the sheet at random. 
We wish to find the probability that the needle 
crosses 0, 1, 2, • • - of the rulings. Figure 6A shows a 
typical result of a trial. Let x be the perpendicular 


CONFIDENTIAL 


16 


PROBABILITY 


distance from the point of the needle to the first rul- 
ing that the needle touches, and let 6 be the angle 
made by the needle with a line parallel to the rulings. 
The number of rulings crossed by the needle is shown 
in the following table : 


If l sin 6 lies between 

x + a and x + 2a 
x and x + a 
x — a and x 
x — 2a and x — a 
x — 3a and x — 2a 


The number of 
rulings crossed is 

2 

1 

0 

1 

2 


Now by “throwing the needle at random” is meant 
simply that all values of x between 0 and a are 
equally likely, and all values of 6 between 0 and 2r are 
equally likely. The distribution functions for x and 6 
are therefore simply 


F 1 


a 



The throwing of the needle is equivalent to choosing 
a point at random in a unit square whose coordinates 
are F i and P 2 . The regions of the square correspond- 
ing to 0, 1, 2, • • • rulings crossed are separated by 
the curves 


x + na = l sin 6 (n = • • • , — 2, — 1, 0, 1, 2 • • •). (5) 


The structure of the square for the special case 
l = 3a is shown in Figure 6B. The probabilities of 
obtaining 0, 1, 2, or 3 crossings may be found analyti- 
cally by integration or graphically by measuring the 
areas on the square distribution diagram. The results 
are shown in the following table : 


No. crossings 
0 
1 
2 
3 


Probability 

0.107 

0.227 

0.314 

0.352 


2- 1 - 4 Compound Probabilities 

If there are two results, A and B , either or both of 
which may arise from a given set of causes, there are 
a number of probabilities which require expression. 
We shall use the following notation : 

P(A) = probability that A occurs if nothing is 
known about B. 


P(B) = probability that B occurs if nothing is 
known about A . 

P(AB) = probability that both A and B occur. 

P(A\B) = probability that A occurs if B is known 
to have occurred. 

P(B\A) = probability that B occurs if A is known 
to have occurred. 

We shall also use the expressions / and $ for “not A” 
and “not B,” so that, for example, P(/|P) is the 
probability that A does not occur, if B does occur. 

Such a system can be represented generally by the 
choice of a random point in a plane area (Figure 7). 



Figure 7. Graphical representation of compound prob- 
abilities. 


This area may be divided into four regions 1 3 corre- 
sponding to the four possible results: AB, / B , A$, 

The ratios of the areas of these regions to the total 
area are the four fundamental probabilities P(AB), 
P(/P), P(A$), and P(fL$)\ Obviously we have 

P(A) = P(AB) + P(A$) , (6) 

P(P) = P(AB) + P(4B ) . 

We must now consider the conditional probabilities 
P(A\B), P(B\A), etc. If B is known to have hap- 
pened, the random point is known to have fallen in 
the combined area AB + /P, but is equally likely 
to be anywhere in this area, while the result A occurs 
if, and only if, the point falls in AB. The probability 
P(A\B) is therefore the ratio of the area AB to the 
area AB + fL B , or 

P { A\B)= - 1(AB) =P(AB)' 

P(AB) + PUB) P{B ) 

Hence 

P{AB) = P{B) ■ P(A \ B) . (7) 

b These are drawn as connected regions in Figure 7, but 
this is not always the case. 


CONFIDENTIAL 


FUNDAMENTAL CONCEPTS 


17 


That is: the probability that A and B both happen is 
the product of the probability that B occurs if noth- 
ing is known about A, and the probability that A 
occurs if B is known to have happened. 

In some cases P(A\B) = P(A) . In this case we say 
that A is independent of B. In terms of the funda- 
mental probabilities P(AB), etc., A is independent 
of B if 

P(A\B) = = P{A) 

P(AB)+P(/B ) 

= P{AB) + P{A$), 

or 

P(AB) = [P(AB)] 2 + P(/B)P(AB) 

+ P(AB)P(A$) + P(A$)P{jiB) 

= P{AB)[P{AB) + PUB ) + P(A$)] 

+ P{A$)PUB) 

= P(AB)[1 - Ptffi] + P(AJl)PtfB) . 

This simplifies to 

P^P\ _ P(A$) _ ( w hen A is independent of B) . (8) 

PUB) PUt) 

It will be noted that the condition that B is indepen- 
dent of A reduces to the same form, i.e., that B is 
independent of A if A is independent of B. 

Interesting and nontrivial examples illustrating the 
general principles of probability theory are very diffi- 
cult to obtain this early in the discussion. Examples 
with tossed coins or dice are simple enough to satisfy 
fairly well the simple mathematical concepts we are 
discussing, but they are a far cry from the practical 
problems we hope to discuss later. On the other hand, 
these practical problems require concepts and meth- 
ods we have not yet discussed in order to solve them, 
or else must be hedged about by so many restrictions, 
in order to fit them to the mathematical principles 
being discussed, that they seem quite artificial. The 
example given next will illustrate the principles of 
compound probability, but will also illustrate the 
difficulties in obtaining examples. 

We suppose a point Pi placed at random some- 
where within a strip of width lOd. In order to make 
the example illustrate the principles we have dis- 
cussed heretofore, we must imagine that the distance 
X\ of Pi from one side of the strip is chosen at ran- 
dom. As a partial connection with practical problems 
which we shall discuss in more detail later, we might 


imagine Pi to be the position of a bomb crater pro- 
duced by a bomber during area bombing. (It would 
be difficult to imagine the sort of area bombing which 
would exactly satisfy the requirements of Pi falling 
exactly inside the strip and being completely at ran- 
dom inside the strip, but it would not be difficult to 
imagine a type of area bombing which would approxi- 
mately satisfy these requirements.) Inside this strip 
are a series of six strips of width d (railroad tracks, 
perhaps) which we are interested in bombing. This is 
shown in Figure 8. The random variable for point Pi 
will then be (aJi/10 d). We can say that, when the 
value of this variable is between 0.3 and 0.4, track 2 
will be destroyed. The probability that this track will 
be destroyed will therefore be the difference between 
these two quantities, which is equal to one-tenth. 

Now suppose another bomb is dropped within the 
strip. The situation relating to the position of this 
point P 2 will depend upon the relationship between 
the two bombs dropped. The second bomb might be 
dropped by a different plane coming over at a differ- 
ent time and having no relation to the first plane. In 
this case we can probably say that the dropping of 
the second bomb is independent of the dropping of 
the first bomb, and the second random variable 
(z 2 /10 d) is independent of the first random variable. 
The square area representing probabilities will then 
be as shown on the left side of Figure 8. The numbers 
in the various small squares indicate the particular 
strip within which the two bombs fall. Since the two 
variables are completely random, the probabilities of 
occurrence are proportional to the areas involved. 
For instance, the probability that the first bomb fall 
on track 2 is one-tenth. The probability that one or 
the other of the bombs fall on track 2 is the area of 
all those rectangles which have a number two inside 
them, i.e., 0.19. 

The definitions discussed earlier in this section can 
also be illustrated. For instance, the probability that 
track 5 will be hit by the second bomb, if we know 
that the first bomb has hit track 2, will be 


P( 5|2) = F ^- 
P( 2) 


0.01 

0.10 


= 0 . 1 . 


This is equal to the probability P( 5) that track 5 is 
hit by the second bomb when we do not know what 
happened to the first bomb. On further analysis it will 
be seen that this simple relationship comes about due 
to the fact that the areas involved in the present case 
are all rectangular, with boundaries parallel to the 


CONFIDENTIAL 


18 


PROBABILITY 



X,,X 2 AT RANDOM IN STRIP, 
X 2 INDEPENDENT OF X, 


X, AT RANDOM INSIDE STRIP, 
AT RANDOM b s 2d 



P( 00) = 0,16 

P(0,n) = P(m,0) = 0.04 

P(m,n) = 0.01 

m,n = 1,2 ,3, 4, 5, 6 
P(m|n) = 0.1 

P(m) = P(n) =0.1 


P (00) = 0.27270 

P(Ol) = P(I0) = P(06) * P(60) = 0.04186 

P(ll) = P(22) = = P(66) = 0.01626 

P(I2) =P(2I) = = P(65) = 0.02007 

P(02) = P(20) = P(I3) = P(3I) = 

-~=P(64) = P(05) = P(50)= 0.02180 
P(m) = P(n) = 0.1 


Figure 8. Example of independent and conditional probabilities. 


CONFIDENTIAL 


FUNDAMENTAL CONCEPTS 


19 


edges of the probability square. This has occurred 
because the two random variables are independent of 
each other. Equation (8) can also be verified in this 
case, and again it is not difficult to see that the equa- 
tion is satisfied because the subareas are rectangular 
in shape with their edges parallel to the main square. 

In contrast, let us consider next that the second 
bomb is dropped a given distance b = 2d away from 
the first bomb in a random direction (this case is re- 
lated to the Buff on needle problem). This is perhaps 
a simplified picture of what happens when two bombs 
are dropped in train. In the actual case, of course, the 
distance is not exactly determined; however, this 
would mean introducing another random variable, 
and so, for the present example, we shall assume that 
the distance between points Pi and P 2 is exactly 2d. 
The two random variables are therefore (xi/10 d) and 
(P/2tt) (shown in Figure 8). We note that, since we 
have required that Pi fall at random within the full 
strip, in this example it sometimes occurs that P 2 will 
fall outside the strip. According to our assumptions, 
however, it can never fall more than a distance 2d 
beyond the edges. 

The probability square for this second case is 
shown at the right in Figure 8. Since the two var- 
iables are not independent, we see that the areas 
corresponding to the different tracks being hit are 
not rectangles and, in fact, that a good many of them 
are missing entirely. For instance, according to our 
assumptions, it is impossible for track 5 to be hit by 
the second bomb if track 2 is hit by the first bomb. 
The probability of the first bomb hitting one of the 
tracks is still one-tenth, and, as might be expected, 
the probability of the second bomb hitting one of the 
tracks, if we do not know what has happened to the 
first bomb, is also equal to one-tenth. The probability 
that two adjacent tracks be hit, such as P(l,2), is 
greater in this case than it was in the previous case, 
and the probability that two tracks a distance 2d 
apart be hit is somewhat larger still. 

The probability that the second bomb will land on 
track 4 if we know that the first bomb has landed on 
track 2 is given by the following equation : 

p(4|2) = 0 02180 = o,2180 . 

P( 2) 0.1 

We see in this case that the result is not equal to 
P(4). To check equation (8) we compute the follow- 
ing quantities: 


P{U) = 1 - [P(20) + P(21) + P(22) + P(23) 
+ P(24) + P(25) + P(26) + P(04) 
+ P(14) + P(34) + P(44) + P(5 4) 
+ P(64) ] 

= 1 - P(20) - P(21) - P( 22) - P( 23) 

- P(24) - P(34) - P(44) - P(54) 

- P(64) 

= 0.82180 ; 

P(24) = P(04) + P(14) + P(34) + P(44) 

+ P(54) + P(64) 

= P(34) + P(44) + P(54) + P(64) 

= 0.07820 
= P(24) . 


A similar computation indicates that equation (8) 
does not hold, and therefore that the position of bomb 
one cannot be independent of the position of bomb 
two. 


Pi 24) 

P(2 4) 


0.279 ; 


P(24) 

P(24) 


0.095 . 


This is only natural, since our assumption regarding 
the fixed value of b makes independence impossible. 
The fact that the position of the second bomb is not 
independent of the position of the first bomb shows 
up in the nonrectangular division of the various areas 
in the probability square and in the corresponding 
impossibility to satisfy equation (8) . 

A number of conclusions which have an approxi- 
mate application to certain practical problems in 
train bombing might be deduced from this example. 
For instance, we see that the probability P(00) is 
larger when the bombs are dropped in train than 
when they are dropped independently. This is nat- 
ural, of course, since if the first bomb misses, the 
second bomb is more likely to miss when it is in train 
than when it is not. However, we will discuss the 
train bombing problem in more detail later. 


2.1.5 Expected Values 

Suppose we have decided on the stochastic variable 
for the problem we are interested in, and suppose our 
analysis has made it possible to determine the func- 
tional relationship between this stochastic variable x 
and the random variable £ which has equal proba- 
bility of being anywhere in the range from zero to 


CONFIDENTIAL 


20 


PROBABILITY 


unity. In addition to knowing the relative probabili- 
ties for the occurrence of different values of x, we 
will often wish to put our expectation of the results of 
a large number of trials in terms of average, or ex- 
pected, values. 

In practice the average value would be obtained by 
making a large number of trials at random and com- 
puting the average value of x from these trials. If we 
have analyzed our problem correctly, we should be 
able to predict the value of this average with more or 



0 2 4 


X 

A f (X)« 0.25 



Figure 9. Examples of distribution functions and prob- 
ability densities with equal expected values of x. Values 
of £ = F occur at random. 


less accuracy. The predicted or idealized value of the 
average will be called the expected value of the sto- 
chastic variable x. The actual average value obtained 
by making a series of trials would differ from this 
expected value by an amount which we would expect 
usually to diminish as the number of trials increases. 
More will be said concerning this later. 

As an example of these general statements, let us 
consider the distribution functions and probability 
densities given in Figure 9. In the first case, the prob- 
ability density is constant, independent of x, so that 


x is directly proportional to the random variable 
F = £. Consequently, x is equally likely to have a 
value anywhere in the range 0 to 4. In a large number 
of trials one would expect to find a value of x larger 
than 2 just as often as a value of x smaller than 2; 
one can see intuitively that the expected value of x, 
which should correspond closely to the average of a 
large number of tries, would equal 2. 

Glancing at the second figure, we note that the 
probability density has a maximum near the center 
of the range for x, and therefore x does not vary 
linearly with the random variable F = £. Neverthe- 
less, in this case also, due to the symmetry of the 
figure, one would intuitively see that the expected 
value of x is again 2. 

What then is the difference in the behavior of the 
two cases? How could we most easily distinguish be- 
tween the two if we did not have the curves for prob- 
ability density in front of us? One sees that in case A 
the value of x, taken from an individual trial, is more 
likely to differ widely from the expected value than is 
the case B. In the first case the probability density is 
uniform, whereas in the second case the probability 
density is largest near x = 2, and falls off to zero at 
the two ends of the range. 

It would be a useful thing to have a numerical 
measure of this chance of large discrepancy of an 
individual trial away from the expected value. The 
average value of the difference between an individual 
trial and the expected value is not a satisfactory 
measure because this, by definition, has positive 
values as often as negative values, and the final aver- 
age should cancel out to zero. If we remove the alge- 
braic sign of the difference, however, by squaring, we 
can obtain a numerical measure. Specifically we com- 
pute the average (or rather, the expected value) of 
the square of the difference between the result of an 
individual trial and the expected value of the result. 
The square root of this average square deviation will 
be called the standard deviation. 

Let us now try to state these concepts in a little 
more precise manner. 

If a very large number of choices of a random var- 
iable is made, we feel intuitively that if the range of 
the variable is divided into any number of equal 
intervals, we will choose values equally often in each 
of the equal intervals. In fact, this is essentially what 
we mean by our definition of a random variable. This 
does not mean that this will be the actual result of a 
trial — on this we shall have more to say later. Never- 
theless, we shall call this the expected result. 


CONFIDENTIAL 


FUNDAMENTAL CONCEPTS 


21 


In particular, if we may make N choices of a ran- 
dom variable £ whose range is 0 to 1, the expected 
number of values in any infinitesimal interval d£ 
is Nd%. 

If a; is a stochastic variable determined by £, the 
average value of x, if the expected result is obtained, 
is called the expected value of x, E(x). This is ob- 
viously given by 


E(x) = / xd£ 


(9) 


= xf{x)dx (if f exists) . 


The standard deviation indicates that the result of a 
single trial differs on the average by a little more than 
a unit on either side of the average value, 2. This is 
often written as E{x) =b a(x ) ; in this case 2 =h 1.16. 

For the case of Figure 9B the expected value and 
the standard deviation turn out to be 


fix) = ~ sin J ; F(x) = ^ 



E{x) =- f\ sin — dx = 2 ; (13) 

8 J o 4 

E{x>) = - / x 2 sin — dx = 4.7578 ; 

Sjo 4 


If there are only a discrete set of values Xi possible for 
x, with probabilities p^ this reduces to 

E(x) = y y y j x i pi. (10) 

i 

The continuous case may be evaluated graphically 
by plotting a; as a function of £. E{x) is then the area 
between the curve and the £ axis. 

It should be noted that the expected value of a sum 
x + y is the sum E(x) + E(y). Naturally the expected 
value of ax is aE(x), if a is a constant. 

According to our previous discussion, we will define 
the standard deviation cr of a stochastic variable to be 
the square root of the expected value of the square 
of the difference between x and E{x). 

a\x)=E{[x-E(x)V} 


c 2 = 0.7578 ; <7 = 0.8705 . 


We notice that the standard deviation a is less for 
this case than for the case of Figure 9A given in 
equation (12). This is to be expected, since the proba- 
bility density of Figure 9B shows a more pronounced 
clustering of values around the expected value 2. 

If a point is chosen at random within a circle of 
radius a, we may find the expected value of the dis- 
tance from the point to the center. For, in this case, 
if x and y are coordinates with origin at the center 
of the circle, 


E(r) 


hff 


rdxdy , 


when the integration is over the circle. Transforming 
to polar coordinates, 


= E{x 2 - 2xE(x) + [E(x)] 2 } 

= E{x 2 ) - 2E{x)E{x) + [E(x)Y 
= E{x 2 ) - [E(x)} 2 . 

In the case shown in Figure 9A, x = 4£, 
E ( 40 = ±f\di; = 2 ; 

E( 16f) = P (1 «p)dt = ~ ; 

J 0 3 


(id 


( 12 ) 


E(r) = — if r 2 drdd 
ira 2 JJ 


2 

-a . 
3 


We also have 

E(r 2 ) = JJ" r 2 • rdrdO 


1 2 

= - a 2 ; 

2 J 


* 2 (4£) 


IT 

3 


12 

~3 



1.155. 




CONFIDENTIAL 


22 


PROBABILITY 


2.2 THE SIMPLE DISTRIBUTION 
LAWS 

With a trial or series of trials involving random ele- 
ments, such as operations of war often turn out to be, 
the result of an individual trial cannot be predicted 
exactly in advance. What can be predicted, if we can 
analyze the problem thoroughly, is the probability 
of certain events occurring, which can be expressed 
in terms of the distribution function or the proba- 
bility density. If the probability of a certain event 
occurring is large, then we can reasonably expect that 
for most of the trials this event will occur; unless the 
probability is unity, however, there is always the 
chance that we will be unlucky in the first or succeed- 
ing tries. 

When a large number of trials can be carried out, a 
knowledge of the distribution function enables one 
to predict average values with more or less precision. 
As more and more trials are made, we can expect the 
average value of the result to correspond closer and 
closer to the expected value which has been discussed 
in the previous section. One can also compute the 
chance that the average result of many trials will 
differ by a specified amount from the computed ex- 
pected value. If the general form of the distribution 
function is known, one can even compute the proba- 
bility that the average results of a second series of 
trials will differ by a specified amount from the aver- 
age result of a first series of trials. 

Such calculations are extremely important in 
studying operations which are repeated many times, 
such as bombing runs or submarine attacks. If the 
first 50 antishipping strikes result in 10 enemy vessels 
sunk, it might be important to compute the proba- 
bility that the next 50 strikes would sink at least 8 
enemy ships. This can be done if the distribution 
corresponding to the attack is known at least ap- 
proximately. 

Consequently, it is important to compute the dis- 
tribution functions for a number of very general 
statistical situations, which correspond more or less 
accurately to actual situations often encountered. In 
a great number of cases this correspondence is not 
exact, but is close enough so that statistical predic- 
tions can be made with reasonable success. The more 
useful cases will be discussed in this section. It should 
be emphasized again that there are many situations 
encountered in practice where none of the common 
distribution laws apply, so that it is not wise to apply 
the results of this section blindly to a new problem. 


2.2.1 Binomial Distribution 

The simplest case is where the result of the trial 
can be called either a success or a failure, such as the 
trial of tossing a coin to get heads, or the firing of a 
torpedo at an enemy vessel. In some of these cases it 
is possible to determine the probability of success at 
each trial; we can call this p. The probability of 
failure in a given trial is therefore q = 1 — p. 

A typical random sequence of successes, S, and 
failures, F, is shown in Table 1 where the probability 
of success p = 0.5. This sequence is typical of random 


Table 1. Random sequence of successes ( S ) and failures ( F ), 
when the probability of success is 0.5. 


FFFSS 

FFSSS 

FSSSS 

SFFSF 

SSSSF 

SFFFF 

SSFSF 

SFFFS 

SFFSS 

FFSFF 

SFFFF 

FSFSF 

FSFFS 

FSSSS 

FFSSS 

SFSSF 

SSSSS 

SSSFF 

FFSFF 

SFFFF 


events and illustrates a number of their properties. 
Other sequences can be obtained from Table I at the 
back of the book. 

In the first place, the average result of a small 
number of trials may give a completely erroneous 
picture of the probability of success of the rest of the 
trials. In this case the first three trials were all fail- 
ures, which might discourage one if it were not known 
that the probability of success is 50 per cent. We no- 
tice also that the seventeenth set of five trials is all 
successes. If this were the first set of five trials, it 
might lead to overconfidence. 

In the twenty sets of five tries each, there is one 
with all five successes, there are three with four suc- 
cesses and one failure, six with three successes and 
two failures, five with two successes and three fail- 
ures, five with one success and four failures, and there 
is none with five failures. It is often useful to be able 
to compute the expected values of the frequency of 
occurrence of such cases. The expected value of the 
fraction of times a given proportion of successes and 
failures occur in a set of trials is, of course, the proba- 
bility of occurrence of the proportion. We shall com- 
pute the probability of occurrence of $ successes and 
n — s failures in a set of n trials, when the probability 
of success in a single trial is p. 

Fully as important is the inverse problem where we 
have made a series of trials and wish to deduce from 
them the probability of success p for an individual 
trial. An examination of Table 1 will indicate that 


CONFIDENTIAL 


THE SIMPLE DISTRIBUTION LAWS 


23 


Table 2. Comparison of results of Table 1 with expected values (n = 5, p = 0.5). 


•* 

s = 5 

s = 4 

s = 3 

s = 2 

s = 1 

s = 0 

Per cent success in five trials 

100 

80 

60 

40 

20 

0 

Fraction of times combination observed 

0.05 

0.15 

0.30 

0.25 

0.25 

0 

Expected value of fraction, P(s, 5) 

0.03 

0.16 

0.31 

0.31 

0.16 

0.03 


Observed mean square deviation (s — 2.5) avg 2 = 1.35 


we cannot compute exactly the value of p from the 
results of a finite number of trials (unless, of course, 
we can analyze the situation completely by mathe- 
matics and predict the value of p) . What we can do is 
to compute the most probable value of p and compute 
the probability that p has other values. However, 
this knowledge is sufficient to enable us to compute 
expected values for another similar series of trials. 
This problem will also be discussed later in the 
section. 

By the law of compound probabilities, if each trial 
is independent the probability of a given sequence of s 
successes and n — s failures in a given order (such 
as FSFFF, for instance, or else FFFFS) is 


E(s ) = ^2 sP(s, n) 


U I 

E n ! 
.9 \ (n — 


s=0 

d 


s ! (n — s ) ! 

1 


psqr 


= p-T 

dp s \{n — s) ! 


ps^ 


= V— (p + lY 

dp 


np(p + q) n ~ 


psqn-s , where q = 1 — p . 


or, since p + q = 1, 


Corresponding to any given values of s and n there 
are 

n\ 

s \{n — s ) ! 

different orders 0 in which the s successes and n — s 
failures can occur (for instance one success and four 
failures is either SFFFF, FSFFF, FFSFF, FFFSF, 
or FFFFS). It follows that the total probability of 
obtaining s successes and n — s failures in n trials is 

P(s , ») = , ■ (! 4 ) 

s ! [n — s) ! 

If we expand (p + q) n by the binomial theorem, we 
see that P(s, n) is just the value of the term contain- 
ing p 8 q n ~ 8 in the expansion. For this reason the dis- 
tribution of the probability of obtaining s successes 
in n trials is known as the binomial distribution. 

The expected number of successes is, by equa- 
tion (10), 

c For a discussion of the laws of permutations and combina- 
tions, see Fry. 6 


E(s) = np . (15) 

In other words, the expected number of successes is 
equal to the number of trials times the probability of 
success per trial, which is as it should be. In the case 
given in Table 1 the expected number of successes in 
five tries would be 2.5. In the sequence of twenty 
sets shown in Table 1, all possible values of s except 
s = 0 occurred. The fractional number of times a 
particular value of s occurred in the sequence of tests 
is given in Table 2. These fractions are also compared 
with their expected values P(s, 5) . The correspondence 
is fairly close. 

The observed value of s, the number of successes 
in five trials, may differ considerably from the ex- 
pected value 2.5. For instance, in five cases out of 
twenty the value is s = 1. This is reflected in the 
value of the mean square deviation computed from 
the actual results given in Table 1. This comes out to 
be 1.35, having a square root approximately equal 
to 1.2. We can express the observations given in 
Table 1 by saying that the number of successes in 
five trials is 2.5 =b 1.2. The value of the root-mean- 


CONFIDENTIAL 


24 


PROBABILITY 


square deviation gives a measure of how widely an 
individual series of trials will deviate from the ex- 
pected value. 

To find the standard deviation, 

s=0 v ' 

= Pj-\p-~(p + ?)”] 

dp L dp J 

= np + n(n — 1 )p 2 . 

HenCe *»(«) =E(s 2 ) - [E(s)V 
= np( 1 - p) 

= npq . (16) 


The standard deviation is, of course, the expected 
value of the root-mean-square deviation. For Table 1 
we have found that the root-mean-square deviation 
was a/ 1-35. The stan dard deviation for this case 
turns out to be \/T. 25, which is a reasonable check. 
Theoretical calculations would therefore have indi- 
cated that the number of successes in five trials would 
be 2.5 ± 1.1, which corresponds fairly closely to the 
actual results of the sequence given in Table 1. 

As an example for the reader, it might be instruc- 
tive to analyze the following random sequence of 
successes and failures for the probability of success 
equal to 0.3 : 


FFFFF 

SSFFF 

FFFFF 

FFSFF 

SFFFF 

SFFFF 

SFSFS 

FSSFS 

SSSFF 

FFFFS 

SFFSF 

FFFFF 

FSSSS 

SFFSF 

FFFSS 

FFSFS 

FFFFF 

FSFSF 

FFFFF 

SFFFF 


Now, suppose we are given the sequence of results 
of Table 1, and are asked to find the value of p, the 
probability of success of an individual trial. This 
question will be discussed more completely in the 
section on sampling, but it is instructive to com- 
mence the discussion here. The most probable value 
of p would be obtained by dividing the number of 
successes actually observed by the total number of 
trials, which for any single set of five trials may differ 
widely from the true value. A crude measure of how 
widely the true value may differ from the observed 
value can be computed by assuming that the value 
of p actually equals the observed value of s/n and 
computing a mean square deviation from this as- 
sumed value of p : 


Rough estimate : a 2 



(17) 


trials. For example, if we performed only the first set 
of five trials in Table 1, we would then estimate that 
the expected number of successes in five future trials 
would be 2 d= 1.1 where the figure after the plus-or- 
minus sign is computed from the expression above: 


V2(l - 0.4) = 1.1 . 


Our estimated value of p from the first set of five 
trials is therefore 0.4 =L 0.2. If we wish to make this 
estimate more accurate, we must perform a larger 
number of trials than five. 

The formula given above for obtaining a rough 
estimate of a 2 breaks down completely in certain 
cases. For instance, in the seventeenth set of trials 
(which turned out to be all successes) the rough esti- 
mate turns out to be zero, since s = n. A more satis- 
factory way of estimating the likely range of p can be 
obtained from equation (14). For an observed num- 
ber of successes s in n trials, we can find out over 
what range of assumed values of p the probability of 
occurrence of this result, P(s, n) is greater than one 
chance in three (or perhaps one chance in ten if one 
wishes to be finicking). If we had been unlucky 
enough to obtain five successes in five trials when the 
“actual” value of p was 0.5, we would not have been 
able to obtain a very good estimate of the value of p 
from only these five trials. All we could have said 
from this one sequence of trials was that there was 
less than one chance in three that the true value of p 
was smaller than 0.8, and that the chances were less 
than one in ten that the true value of p was less than 
0.6. The difficulties are inherent in the situation; five 
trials are too few to yield a dependable value of p. 

An instructive illustration of these general state- 
ments lies in the criticism of an occasionally used pro- 
cedure for determining the percentage of duds in a 
batch of shells (or torpedoes or grenades) : to fire the 
shells until one dud appears, and then to stop the 
test. Suppose n — 1 shells were fired before a dud 
appeared and then the nth shell was a dud. The pre- 
dicted fraction of duds, based on such a test, would 
be 1 /n and the predicted number of duds in A shells 
would be N /n. 

But it is rather dangerous to base predictions on 
the observation of only one failure; we could be quite 
seriously off in our prediction of how many duds 
would be in the next n shells. From equation (14), the 
probability of finding one dud in n trials, when the 
expected fraction of duds is q is 


where s is the observed number of successes in n 


P{n — 1, n) = nqi 1 — q) n 1 , 


CONFIDENTIAL 


THE SIMPLE DISTRIBUTION LAWS 


25 


which approaches nqe~ nq when q is small. In this case 
we do not know q , but we wish to determine the range 
of values of q over which the probability nqe~ nq has 
reasonably large values (is larger than 0.1 for in- 
stance) . 

The maximum value of P(n — 1, n) is e~ x , corre- 
sponding to the most probable value for q of l/n. In 
other words, the most probable prediction from our 
series of n trials is that there is one dud in every n 
shells. But, if we assume that q is twice this (2 duds 
per n shells), P(n — 1, n) is 2e~ 2 , which is still larger 
than 0.1. In fact, the range of values of q for which 
P(n — 1, n) is larger than 0.1 (i.e., for which the 
result of our trials would be reasonably probable) is 
from approximately 0.11 /n to 3.5 /n. Therefore, 
it is reasonably probable that the “most probable” 
value of the fraction of duds, l/n, is nine times 
larger than the “correct” value, or is too small by a 
factor of nearly 1+. In other words, it is fairly likely 
that the next n shells would have four duds instead 
of one; it is also likely that there would be only one 
dud in the next 9 n shells. 

The moral of this analysis is that, if we wish to be 
“reasonably certain” of the fraction of duds in a lot 
of shells, we must fire enough shells so that more than 
one dud appears (in practice, enough trials so that at 
least ten duds appear is adequate) . 

A much more thoroughgoing analysis of these ques- 
tions is given later in this chapter. 

We are frequently interested in not the probability 
of obtaining exactly $ successes, but rather a number 
of successes between two limits, Si and s 2 . When n, 
«i, and s 2 are large, the calculation of the individual 
probabilities for all the values of s between Si and s 2 
becomes very laborious. These calculations can be 
simplified by the use of summation formulas based 
on the beta function, which we shall now derive. 

The probability that in n trials we obtain s or 
fewer successes is 

8 , 

E n ! 

i=o kl(n-k)\ Pq 


The derivative of P(^ s, n) with respect to p (re- 
membering q = 1 — p) is easily seen to be 

dPig^n) = _ n(n - 1) ■ ■ • (n - s) t 

dp s ! 

n ! 

= — V s Q n 8—1 

s\(n-s- 1)! 

The other terms in the sum all cancel. Hence 
C v n T 

P(gs,n) = - — — p s ( 1 - p^-’-'dp + c. 

Jos \{n — s— 1) ! 

But, if p = 1, obviously P(^s,ri) = 0. Hence 

f 1 

c = +1 — — p‘( i - py-'-'dp , 

Jo s !(n — s — 1) ! 

and 

C 1 n T 

P( £ s,n) = / : — - p*(l - p)”-*- 1 dp . 

Now the incomplete beta function is defined as 

B x (a, b ) =^* p° _1 (l - p) 6_1 dp , (19) 

and the complete beta function as 

B{a, b) = /VKl-p)^p = ( °~i ) , !(b ~ 1 ?, ! • ( 20 ) 

Jo (a + b — 1) ! 

We therefore see that the distribution function for 
this case is 

F b (s, n) = P(^s, n) 

- 1 _ B P ( S + 1 ,n — s) 

B(s + 1, n — s) 

= 1 - I p (s + 1, n - s) , (21) 


, n . . n(n — 1) . „ 

= g n + — p? n_1 + — —j — " pY 2 • • * 


. n(n — 1) • • • (n — s + 1) w a 

-j- L 1 ! — ' psqn-s 

s ! 


(18) 


where Fb(s, n) is the binomial distribution function, 
that is, the probability of obtaining s or fewer suc- 
cesses in n trials. 

Tables of the ratio 


Ua, b) 


B x (a, b) 
B(a, b) 


CONFIDENTIAL 


26 


PROBABILITY 




Figure 10. A. Binomial distribution function Fb(s, n ) 
for n = 100, p = 0.1. B. Probability Fb(s — 1, n — 1) 
that n or more tries are required to obtain 10 hits, for 
s = 10, p = 0.1. 


is 1/10 X 100 = 10. The standard deviation is given 
by 

= 100 X — X — = 9 , 

10 10 

or 

<7=3. 

These two results are sometimes summarized by 
saying that the expected number of hits is 10 ± 3. 
The probability P(^s, 100), or F b (s, 100), of obtain- 
ing s or fewer hits is shown in Figure 10A. 

We may also use these same results to determine 
how many trials will be needed to obtain a given 
number of successes, for the probability that n or 
more trials will be needed to obtain $ successes is 
exactly the same as the probability that n — 1 trials 
produce s — 1 or less successes. Hence, using obvious 
notation, 

P(s, ^ n) = P(^ s — 1, n — 1) 

= F b (s — 1 , n — 1 ) 

= 1 — I p (s, n — s — 1) . (23) 

In the example of the gun, if 10 hits are required, the 
probability that n or more shots are required is 
shown in Figure 10B. 


have been published 6 and serve as the most con- 
venient method of evaluating F b (s, n). A short table 
of F b (s, n) is given at the back of the book (Table IV). 

With F b (s, n ) known, the probability that the 
number of successes in n trials is between Si and s 2 is 
easily found. In fact 


P(si^s^s 2 , n) = 


k \{n — k ) ! 


= P(^s 2 , n) - P(£ Sl — 1, n) 


2.2.2 The Normal Distribution 

When the number of trials is large, the frequency of 
successes in a series of repeated trials becomes prac- 
tically a continuous variable. Instead of s, it then be- 
comes more convenient to use x = s/n as a new 
variable. The expected value of x is then p, and its 
standard deviation is given by 

n 


= F b (s 2 , n) — F b {si — 1, n) 

-Ip(Si, TL S] | 1) I p(s 2 ~]~ 1, Ti S2) » 

( 22 ) 

To illustrate these results, suppose that a gun has 
a probability of 1/10 of hitting a target on each shot. 
If 100 rounds are fired, the expected number of hits 


The probability that the fraction of trials resulting 
in success is less than x is of course equal to the proba- 
bility that the number of successes is less than xn, 
so that 

P(< x, n) = 1 — I p [nx, n{l — x)] , 

if we neglect terms of the order of unity in compari- 
son with n. 


CONFIDENTIAL 


THE SIMPLE DISTRIBUTION LAWS 


27 



DISTRIBUTION FUNCTION F n — 

Figuke 11 . Normal distribution function F n (y). Ex- 
pected value of y is E{y) = 0, and standard deviation 
a{y) = 1. See Table V at back of book. 


It is sometimes convenient to use still another 
variable y, defined by 



whose expected value is 0, and whose standard de- 
viation is 1 . In terms of y 


f.( y) 


= •P(<2/,°°) 



e u * /2 du ; 


Fn(-y) = 1 - F»(y) . 


(24) 


The curve of y as a function of F is shown in Figure 
11, and a table of values is given in Table V at the 
back of the book. By its definition, F is the random 
variable corresponding to the stochastic variable y. 

The normal distribution law is much used (in 
fact too much used) as an approximation to other dis- 
tribution laws. It is applied, for example, not only to 
long series of repeated trials, but also to fairly short 
series, and also to represent the distribution of un- 
analyzed errors which occur in physical measure- 
ments. Its advantage is that if x is any stochastic 
variable whose expected value is m and standard 
deviation is a, we may define a variable 


a 


and assume for better or worse that y follows the 
normal distribution law. We thus set up a distribu- 
tion law on the scanty basis of only the two constants 
m and a. This procedure, however, is dangerous and 
can lead to very erroneous conclusions unless tests 
are applied to verify the normality of the distribu- 
tion. Nevertheless, the normal distribution is most 
valuable because of its simplicity. 

A number of features of the normal law are obvious 
from Figure 11. The distribution is symmetrical in 
the sense that the probability that the value of y lies 
between yi and y 2 is the same as the probability that 
it lies between —y 2 and —y\. Small values of y are 
more likely than large values. In fact, there is a 50 
per cent probability that y lies between —0.67 and 
+0.67 and a 90 per cent probability that y lies be- 
tween — 1.64 and +1.64. By definition, for a normal 
distribution 


As n becomes larger and larger, the curves of 
P(< y,ri) against y approach a limiting curve, which 
is generally known as the normal distribution curve. 

It is shown in standard works on probability that 
the limiting curve has for its equation : 


E{y) = 0 ; a 2 (y) = 1 . (26) 

Table III at the back of the book gives typical se- 
quences of random values of y and of y 2 . They have 
been obtained by considering the random numbers of 
Table I as five-digit decimal fractions, equal to ran- 
dom values of F n (y ) . From these, by use of tables of y 
as a function of F n , we obtain corresponding values 


CONFIDENTIAL 


28 


PROBABILITY 


of y, the stochastic variable. A constant amount has 
been added to each group of values of y so that the 
average value of y for each group is exactly zero. This 
would not be strictly true for random values of y , 
but it makes the table more useful for some of the 
applications discussed in a later chapter. Nor is it 
true that the actual values of the mean square 
deviation, (y 2 ), for each group are equal to unity, 
the standard deviation. The larger the sample, how- 
ever, the nearer will this be true (for instance, the 
mean square deviation for the whole of Table III 
is 1.015). 

A glance at Table III shows that magnitudes of y 
smaller than unity are fairly common; magnitudes 
larger than two are quite uncommon. This is typical 
of normal distributions. Some interesting and useful 
applications of Table III will be given in Chapter 6. 

Deviations from the point-of-aim of aircraft bombs 
usually follow the normal distribution, with a stan- 
dard deviation in range (along the track of the plane) 
greater than the standard deviation in deflection 
(perpendicular to the track of the plane). Therefore, 
a simple example would be the case of the bombing 
of a carrier, when the plane approaches on the beam. 
In this case the length of the carrier is considerably 
larger than the deflection error, so that misses are 
over or under (i.e., in range) rather than right or 
left, and the problem becomes a one-dimensional 
case. If the standard error of the bombardier and 
bomb in range is <j, and if the width of the carrier is a, 
then the probability of hitting the carrier with a 
single-bomb drop is 


The case where more than one bomb is dropped is 
discussed in Chapter 6. 

2.2.3 The Poisson Distribution 

In our discussions so far of random points on a line, 
we have considered only the case in which the length 
of the line is finite. If the line is allowed to increase in 
length without limit, the probability that a given 
point falls in any fixed interval obviously approaches 
zero. If, however, we choose not one, but a number of 
points, and let this number grow larger in proportion 
to the length of the line, then the probability of find- 
ing any given number of points in any fixed interval 
may be expected to approach a finite limit. 

Suppose that on a line of length L, kL points are 
chosen on the line independently and at random. This 
probability that any one of these points lies in a given 
interval of length x is 

x 

L’ 

and, by the binomial distribution law, the proba- 
bility that exactly m of the kL points will be found in 
the interval x is 

(kL ) ! ( x \ m (i x \ kL ~ m . 
m \(kL — m ) ! \L/ V l) 

as L approaches infinity it is easily seen that this 
approaches 

nig-hx 

m ! 

The expected value of m is 





If the bombardier is poorly trained (i.e., the error a 
is much larger than a) then halving the error will 
double the expected number of hits. On the other 
hand, if the bombardier is good (i.e., a is much smaller 
than a), then a further reduction of error will not 
produce a proportional increase in the number of hits. 
This is another illustration of the general rule that it 
pays more to improve the accuracy of the poorest in 
the team rather than to improve still further that of 
the best. 


E = E{m) = 


m(kxy 


m =0 

= kx . 


(27) 


In view of this result we may write the probability 
of obtaining m points as 


P(m, E) = . (28) 

m ! 

The Poisson distribution occurs under more general 
conditions than the foregoing derivation would indi- 
cate. It may be, for example, that points are not dis- 
tributed uniformly along a line, but with a density 
p(x), where x is now a coordinate measured along the 


CONFIDENTIAL 


THE SIMPLE DISTRIBUTION LAWS 


29 


line. In this case the expected number of points 
falling in the interval (zi, x 2 ) is 


rx2 

E = / p(x)dx . 
J XI 


With this value of E, the probability that m points 
fall in this interval is still given by (28) . To show this, 
let us introduce a new coordinate y, defined by 



and change the scale along the line so that y is uni- 
form instead of x. Then on this distorted line the 
points are also distributed uniformly, so that the 
expected number in the interval (yi, y 2 ) is equal to 
y 2 — 2/i, that is, to the length of the interval. Hence 


the Poisson law holds on the distorted line, and since 
the transformation from x to y is single valued, it 
must have held on the original line. 

It is not even necessary to confine the Poisson law 
to the distribution of points on a line. If points are 
independently distributed over a plane, or through a 
volume, in such a way that the probability of any 
particular point falling in any given region is small, 
then the Poisson distribution holds in the form of 
equation (28) . This result shows that the probability 
of m points being in an interval depends only on the 
expected number, and nothing else. This equation is 
the basis of the Poisson distribution. 

The expected value of m 2 is 

S TP.mp — E 

=E> + E, 

m ! 

m =0 


Table 3. Random sequence of one hundred numbers between 000 and 999, typical of the behavior of a random variable. 
From Table I (page 153). 


577 

131 

608 

360 

359 

716 

352 

423 

386 

032 

737 

646 

257 

939 

736 

701 

646 

934 

337 

661 

170 

680 

634 

089 

318 

533 

398 

720 

077 

228 

432 

338 

255 

586 

415 

263 

806 

838 

393 

745 

059 

699 

586 

193 

784 

663 

983 

274 

171 

141 

355 

327 

648 

592 

760 

094 

129 

790 

187 

556 

303 

146 

673 

734 

807 

552 

669 

753 

417 

110 

640 

430 

737 

170 

346 

205 

491 

217 

187 

733 

000 

182 

328 

947 

028 

557 

192 

510 

550 

541 

870 

025 

984 

851 

293 

313 

557 

384 

286 

960 


These same hundred numbers shown in order of increasing size, to show fluctuating behavior of successive differences. 


000 

025 

028 

032 

059 

077 

089 

094 

110 

129 

131 

141 

146 

170 

170 

171 
182 
187 
187 
192 


25 

3 

4 

27 

18 

12 

5 

16 

19 

2 

10 

5 

24 

0 

1 

11 

5 

0 

5 

1 


193 

205 

217 

228 

255 

257 

263 

274 

286 

293 

303 

313 

318 

327 

328 

337 

338 
346 
352 
355 


12 

12 

11 

27 

2 

6 

11 

12 

7 

10 

10 

5 
9 
1 
9 
1 

8 

6 

3 

4 


359 

360 
384 
386 
393 
398 
415 
417 
423 
430 
432 
491 
510 
533 
541 
550 
552 

556 

557 
557 


1 

24 

2 

7 

5 
17 

2 

6 

7 
2 

59 

19 
23 

8 
9 
2 
4 
1 
0 

20 


577 

586 

586 

592 

608 

634 

640 

646 

646 

648 

661 

663 

669 

673 

680 

699 

701 

716 

720 

733 


9 

0 

6 

16 

26 

6 

6 

0 

2 

13 

2 

6 

4 

7 

19 

2 

15 

4 

13 

1 


734 

736 

737 
737 
745 
753 
760 
784 
790 
806 
807 
838 
851 
870 
934 
939 
947 
960 

983 

984 


2 

1 

0 

8 

8 

7 
24 

6 

16 

1 

31 

13 

19 

64 

5 

8 
13 
23 

1 


Mean value of random variable = 471.3 
Mean value of difference = 10.00 


CONFIDENTIAL 


30 


PROBABILITY 


and the standard deviation is given by 

a 2 = E(m 2 ) - E 2 (m) = E 2 + E - E 2 = E . (29) 

An important property of the Poisson distribution is 
expressed by this equation: the standard deviation 
equals the square root of the expected number. If we 
choose an interval small enough so that the expected 
number E in the interval is one or two, samples con- 
taining zero or 2 E will be frequent (i.e., <r~l). If the 
interval is large enough to expect a hundred, then the 
usual fluctuations about this expected value will be of 
the order of ten; the percentage fluctuation decreasing 
as the expected value increases. 

As an example of the Poisson distribution we can 
analyze Table 3. 

One hundred points on a line of one thousand units 
corresponds to a large enough sample so that the 
Poisson distribution should hold reasonably well. The 
second part of the table shows the distribution of 
these points along a line, as discussed in this sub- 
section. We note the seeming tendency to “ bunching’’ 
which is always evidenced by random events. 

If we count up the number of intervals of ten units 
length (000 to 009, 010 to 019, • • • , 990 to 999) 
which contain no point, we find that 34 of them are 
so characterized (for instance 010 to 019, 040 to 049 
• • • contain no point) ; we find 44 contain one point, 
25 two points, and so on. There are one hundred 
points and one hundred intervals, so the expected 
number of points in an interval is unity. We can 
therefore compare the fraction of intervals having m 
points with the probability P(m, 1) given in equa- 
tion (28) : 

m, No. points in interval 0 1 

Computed probability, P(m, 1) 0.37 0.37 

Observed fraction of cases 0.34 0.44 


are found in an interval equal to or larger than one 
which would be expected to have E points. This 
duality between m and E is another peculiar property 
of the Poisson distribution. Values of the function 
F p (m, E), for various values of m and E, are given in 
Table VI at the back of the book. 

The Poisson distribution will apply in a very large 
number of important situations. It is particularly 
common when the variable x is time. For example, 
the number of alpha particles emitted by a radium 
preparation in a given time interval follows the 
Poisson law, because the particles are emitted inde- 
pendently and at random times. The number of tele- 
phone calls received at a large exchange is also nearly 
random over short intervals of time, and the Poisson 
law again applies. 

This distribution is also useful in studying prob- 
lems of aerial search (see Division 6, Volume 2B). 
If N enemy units are distributed at random over a 
region of the ocean of area A, and if a plane can 
search over Q square miles of ocean per hour of flight, 
then the expected number of units sighted for a flight 
of T hours is 

e _nqt 

A 

In actual practice the enemy units are not usually 
distributed at random, each independent of the posi- 
tion of the other, but in many cases (such as for 
the search for submarines) the results are sufficiently 
similar to those for the Poisson distribution to make 
a study of this distribution profitable. 

For instance, suppose that the expected number of 
enemy units sighted is S per hour of flight, and sup- 

2 3 4 5 

0.18 0.06 0.015 0.003 

0.15 0.04 0.01 0.02 


which is a fairly satisfactory correspondence. 

The distribution function for the Poisson distribu- 
tion is the probability that m points or fewer are in 
the interval : 


F p (m, E) 


m 

x>' E) -X 


P(m,x)dx , (30) 


where P(m, E) is given in equation (28). This inter- 
esting relationship shows that the probability that m 
points or fewer are found in an interval with expected 
number E is equal to the probability that m points 


pose that the maximum range of the plane used is 6 
hours, with maximum load of gasoline. For purposes 
of illustration of the method of analysis, we will 
assume that the plane is supposed to attack each 
unit it sees with one bomb, and that each bomb 
weighs the equivalent of an hour’s worth of gasoline 
(i.e., the plane with 5 bombs could only fly for 1 
hour, and a plane with 2 bombs could fly for 4 hours, 
etc.). Once this extremely simplified case has been 
discussed, it will not be difficult to find methods for 
handling more complicated cases which accord more 
closely with real conditions. 


CONFIDENTIAL 


THE SIMPLE DISTRIBUTION LAWS 


31 


If the plane carries M bombs, the expected num- 
ber of sightings per flight is E = $(6 — M), and the 
probability that* the plane sights m units per flight is 

— [5(6 - il/)]”*e s(M_6) = P[m,S(6 - M)] . 
m ! 

If m is less than M, all m units are bombed, but if m 
is larger than M, only M units are bombed because 
the plane has only M bombs along. The problem is to 
determine the value of M so that, on the average, the 
greatest number of enemy units will be bombed per 
flight. 

One could approach the problem from a naive point 
of view, assuming that the plane always made the 
expected number of sightings per flight. In this case 
the number of bombs M should equal the expected 
number of sightings S( 6 — M), so that M should be 
the nearest integer to [ 6$/(l + S) ] . This result turns 
out to be nearly the correct one, except when S is 
small. When the expected number of sightings per 
6-hour flight is less than 2 (S < 1/3) the simple 
formula would indicate that only one bomb should be 
carried. This naive reasoning, however, neglects the 
fact that there is a chance that more than one unit 
will be seen during a flight, and, if only one bomb is 
carried, this extra chance will be lost. 

To appraise this possibility in a quantitative man- 
ner, we use the Poisson distribution to compute the 
average, or expected, value of the number of bombs 
dropped per flight : 

M OO 

B = ^2nP[n,S(6-M)]+M ^ P[n, 5(6-21/)] 

n = 0 n=M + 1 

M 

= 21/ - U (21/ - n)P[n, S( 6 - 21/) ] . 

n =0 


We see that the naive reasoning discussed above is 
good enough for S = 2 or 1, for the values of M giv- 
ing the largest expected value of B are 4 and 3, re- 
spectively, which are the values given by the simple 
formula [ 6$/(l + S) ] . But for S less than unity the 
effect mentioned above comes more strongly into 
play, and it often turns out that it is best to carry 
more bombs than the simple formula would require, 
just to take advantage of the occasional times the 
plane encounters more enemy units than the expected 
number. For S = 0.6 we should carry 3 bombs in- 
stead of 2, and for S = 0.3 we should carry 2 bombs 
instead of 1 . 

As a somewhat more complicated example, let us 
consider the case of a newsboy who is required to buy 
his papers at 2 cents and sell them at 3 cents, and is 
not allowed to return his unsold papers. He has found 
by experience that he has on the average 10 cus- 
tomers a day, and that customers appear at random. 
How many papers should he buy? 

By “at random,” it is here meant that, in the first 
place, the newsboy has no regular customers, who 
can be counted on to appear regularly, and, secondly, 
that, as people pass him on the street, one person is as 
likely to buy as the next. Under these conditions we 
may expect the Poisson law to hold. 

Now suppose that the newsboy buys Jc papers, and 
that m customers appear. If m is equal to or less than 
k, m papers are sold. The newsboy’s profit is then 
3 m — 2k. If m is greater than k, only k papers can be 
sold, and his profit is exactly k. His expected profit 
then is 


E k 


k 

N ( 3m - 2k ) 

m =0 


10 m e~ 10 
m ! 


00 

+ 1> 

m =&+l 


10 w e~ 10 
m ! 


It is easily seen that 


Values of B for different values of S and M are given 
in Table 4 : 


E/c+i — E k 


k 

=Z(-2) 

m = 0 


10 m <r 10 
m ! 


oo 

+ z 

m =&+l 


10 m e -10 
m ! 


Table 4. Expected number of enemy units bombed 
per flight B for different values of S and of M. 


s = 

2 

1 

0.6 

0.3 

M = 1 

1.00 

0.99 

0.95 

0.78 

2 

2.00 

1.89 

1.60 

1.04 

3 

2.92 

2.33 

1.64 

0.88 

4 

3.22 

1.92 

1.19 

0.60 

5 

1.98 

1.01 

0.60 

0.30 


But, since 


t 

m = 0 


10 m e~ 10 
m ! 


= 1 , 


this may be written 


k 

Ek+i — Ek =1 — 3 ^ ^ 

m =0 


10 m e~ 10 
m ! 


CONFIDENTIAL 


32 


PROBABILITY 


If we imagine the newsboy buying his papers one by 
one, then if he has already bought k papers, he should 
buy the ( k + l)st only if E k +\ — E k is positive. The 
number he should buy is therefore the lowest number 
A; for which E k+ 1 — E k is negative. Table 5 shows the 
calculation in detail. 


Table 5. The newsboy problem. 


k 

10*e“ 10 

k\ 

^ 10 » 6 -io 
m=0 ml 

t?3 

+ 

1 

133 

E k 

0 

0.00005 

0.00005 

0.99985 

0 

1 

0.0005 

0.0005 

0.9985 

0.9999 

2 

0.0023 

0.0028 

0.9916 

1.9984 

3 

0.0076 

0.0104 

0.9688 

2.9900 

4 

0.0189 

0.0293 

0.9124 

3.9589 

5 

0.0378 

0.0671 

0.7987 

4.8715 

6 

0.0631 

0.1302 

0.6094 

5.6697 

7 

0.0901 

0.2203 

0.3394 

6.2784 

8 

0.1126 

0.3329 

0.0013 

6.6088 

9 

0.1251 

0.4580 

-0.3737 

6.6195 

10 

0.1251 

0.5831 

-0.7490 

6.2485 

11 

0.1137 

0.6968 

-1.0904 

5.4962 

12 

0.0948 

0.7916 

-1.3748 

4.4058 

13 

0.0729 

0.8645 

-1.5935 

3.0310 


The first column gives the values of k ; the second, the 
10 m e -10 

values of for m = k; the third, the values of 

m ! 

10 w e~ 10 

— ; the fourth, the values of E k+1 — E k : and 

A m ! 

m= 0 

the last column, the values of E k . The table shows 
clearly that the newsboy should buy only 9 papers, 
and that his expected profit is 6.6 cents. If he made 
the obvious purchase of 10 papers, his expected profit 
would be 6 per cent less. In this, the losses he would 
incur when fewer than the expected 10 customers 
buy more than offset his gains if 10 or more cus- 
tomers come along. The two examples show the pos- 
sible errors of the “naive’ ’ point of view, and indicate 
how the distribution function can be used to obtain a 
better answer. 

23 SAMPLING 

Suppose that a gun has been fired at a target 100 
times, and that 40 hits were obtained. We wish to 
make the “best estimate” of the probability p that 
another shot fired from this gun under the same con- 
ditions will be a hit. We commenced discussing this 
question earlier in this chapter. Now we are better 
equipped to treat it in detail. 


The crux of this problem lies in the interpretation 
of the expression “best estimate.” The difficulty 
arises because of the fact that no matter what the 
value of p may be (except 0 or 1) it is possible that 40 
hits will result in 100 shots. It is therefore impossible 
from the given facts to deduce the exact value of p. 
Any formula which expresses the value of p in terms 
of the number of hits and misses is subject to error. 
All we can calculate is the probability that p will 
have some given value. 

In spite of this difficulty, we feel intuitively that 
the value of p is “probably somewhere around” 0.40. 
That is to say, we are quite sure that p is not 0.01 
or 0.99, although we wouldn’t be prepared to deny 
that the value is not 0.39 or 0.41. In other words, we 
might say that 0.01 and 0.99 are “unreasonable” 
values of p, while 0.39 and 0.41 are “reasonable” 
values. If we are asked why we feel that 0.01 is an 
unreasonable value of p, we might point out that the 
probability of getting 40 hits in 100 shots with 
p = 0.01 is, from equation (14), 

- 100 ! - (0.01) 40 (0.99) 60 , 

40! 60! 

which is about 10~ 52 , and is so small that we can 
“reasonably” assume that such an improbable event 
has not taken place. But, even if we take p = 0.40, 
the probability of obtaining exactly 40 hits in 100 
shots is 

100 -(0.40)*° (0.60)", 

40! 60! 

which is only 0.08. It is not immediately obvious that 
this is large enough to make 0.40 a “reasonable” 
value of p. 

In order to obtain a better criterion of “reason- 
ableness,” or “goodness of fit,” it has become usual 
to adopt a method suggested by Pearson. This 
method does not aim at obtaining a definite value of 
p from the trials (as we have seen, this is not pos- 
sible), but rather seeks to determine a range of values 
of p within which it is “reasonable” to find its real 
value. To test an assumed value of p, we compute 
just the consequent expected result of the experi- 
ment (in this case the expected number of hits, 
100 • p). The agreement between the actual result 
and the expected result is measured by the absolute 
value of the difference between the two, in this case 
1 40 — (100p)|. We now compute the probability 
that in a second experiment, similar to the original, 



CONFIDENTIAL 


SAMPLING 


33 


we would obtain a result which is as far or farther 
from agreeing with the expected result as the actual 
result of the first experiment differs from this ex- 
pected result. We thus get a number which is equal 
to 1 if the first experiment gave exactly the expected 
result, but otherwise is less than 1. This number is 
taken as a measure of the “reasonableness” of the 
value of p tested, and, if it is too small (usually less 
than 0.05), the value is called “unreasonable.” 

When the sample is fairly large, this calculation 
may be simplified by using the normal law as an ap- 
proximation to the binomial law. To illustrate the 
process, let us calculate the “reasonableness” of any 
value p in the case of the gun. The expected number 
of hits is then lOOp, and the difference between the 
observed and expected hits is | lOOp — 40 1 . If we 
shot another 100 rounds, the agreement with the ex- 
pected number of hits would be as bad or worse if the 
number of hits was equal to or more than lOOp 
+ 1 lOOp — 40 1 , or if it was equal to or less than 
lOOp — | lOOp — 40 1 . If we approximate the actual 
distribution of the number of hits in the second 100 
rounds by a normal distribution with a mean lOOp 
and a standard deviation 

c t = "\/ 100p(l — p), [see equation (16)] 

then the probability that the second series gives a 
worse agreement than the first series is 

e~ ix2 dx . [see equation (24)] 

|40-100p|/<r 

The values of this integral are easily obtained from 
Table Y in the back of the book or in standard 
reference books. 7 ’ 8 In the general case where m suc- 
cesses have been obtained in n trials this becomes 

VI /’ (31) 

| to — np\/a 

where a is equal to Vwp( 1 — p) . 

Plots of this “goodness of fit” against the assumed 
value of p for the cases n = 100, m = 40 and n = 10, 
m = 4 are shown in Figure 12. If we take 0.05 as the 
limit of reasonableness, then, for the case n = 100, 
m = 40, the values of p between 0.31 and 0.50 are 
“reasonable” values. In the case n = 10, m = 4, the 
values between 0.16 and 0.69 are “reasonable” values 
of p. It may be pointed out here that for such a small 
sample the normal law is a poor approximation to the 


binomial distribution. Nevertheless, in this case the 
range of “reasonable” values of p is so large that for 
most purposes it would be necessary to make further 
trials before acting on this result, while, in the few 
cases where even the vague knowledge given by the 
small sample is sufficient, the additional vagueness 
added by the use of the normal law can hardly be 
enough to influence the result. Thus the best answer 
to our original question is that the probability p that 



Figure 12. Reasonableness of estimate of probability of 
success p when 10 trials have resulted in 4 successes, and 
when 100 trials have resulted in 40 successes. 


the next shot fired from the gun hit the target is most 
likely equal to 0.40, but it could reasonably have a 
value between 0.31 and 0.5. 

There is one serious disadvantage to this method of 
testing trial values of a probability: the method 
affords no way of taking into account any knowledge 
we may have possessed before the trials which might 
have made one value of p more likely than another. 
If, for example, we knew that the gun being fired was 
one of a lot all manufactured together in exactly the 
same way, and that previous trials on the other guns 
of the lot had all given values of p near 0.3, then it is 
obvious that in the situation of Figure 12 the value 
0.3 is more “reasonable” than the value 0.5, even 
though the curves show these values as equally rea- 
sonable. In most applications, however, we have no 
such information, and, although there exists a real 
logical difficulty with the method, it is ordinarily 
safe to ignore it. 

2.3.1 The x 2 (Chi-Squared) Test 

A great number of trials result in more than just 
success or failure. For instance, a shot from a gun 
may hit the bull’s-eye or the first or second ring as 



CONFIDENTIAL 


34 


PROBABILITY 


well as miss the target entirely. Similarly, a torpedo 
may miss the ship, may damage it, or may sink it. If 
we know the geometry of the problem completely, 
we sometimes may be able to compute the a priori 
probability pi that the fth possibility occur when a 
trial is made, for instance, pi could be the probability 
of hitting the bull’s-eye, whereas p 2 would be the 
probability of hitting inside the first ring, and so on. 
Or, to take another example, probability pi could be 
the probability of shooting down an incoming plane 
with a 5-inch antiaircraft battery when the plane is 
between 6,000 and 4,000 yards away from the battery 
(the guns opened up at 6,000 yards’ range), p 2 the 
probability that the plane is shot down when the 
range is between 4,000 and 2,000 yards, and p 3 is the 
probability of shooting the plane down when the 
range is less than 2,000 yards. 

To generalize from these examples, we can say that 
a given trial may result in a number of different spe- 
cific events, such as hitting the bull’s-eye or the first 
ring, and so forth. Suppose there are s different 
specific possibilities. We can usually choose these 
possibilities in a number of different ways so that the 
value of the integer s will vary according to the na- 
ture of the trial and the degree of detail with which 
we wish to study the results. The quantity s is 
usually called the “number of degrees of freedom” 
of the trials. In order to complete the enumeration of 
the results, we must always include the negative re- 
sults in addition to the s different specific results 
which may come from a trial, that is, we may also 
obtain none of these specified results. In other words, 
it is always possible for the bullet to hit neither the 
bull’s-eye nor any of the rings, but to miss the target 
entirely. In the case of a die, not only may the faces 
1, 2, 3, 4, or 5 turn up, but none of these (i.e., the face 
6) may turn up. In other words, the total number of 
possible results for the trial turn out to be 1 plus the 
number of degrees of freedom, that is, s + 1. 

Corresponding to each possible result there is an 
a priori probability p h p 2 , • • • , p^ • • • , p s+ 1 , where 
the sum of all these probabilities must equal unity. 
If now we make n trials, the expected number of 
trials which result in condition 1 will be np h and so 
forth. The sum of all these expected values must 
equal n. 

But the case we are considering at present is the 
reverse of this. We have just made n trials and we 
wish to find from them “reasonable” values for the 
probability of occurrence of each of the different 
results. In the n trials, mi trials have resulted in 


occurrence 1, m 2 have resulted in occurrence 2, etc. 
The sum of all the m’s must equal n. From this we 
wish to deduce reasonable values of the probabilities 
Pi • 

To be more precise, we wish to know whether the 
observed result is reasonable on the hypothesis that 
the probability that a single trial falls into the fth 
group is pi(2pi = 1). This judgment can be made by 
calculating the probability that a series of n trials 
with the given probabilities would give a result which 
deviates as much or more from the expected result as 
does the observed result. The principal difficulty 
here is in the question of when one result deviates 
more from the expected result than another. For 
example, consider the following results: 


Table 6 


Group 

1 

2 

3 

Expected No. 

3 

12 

4 

Trial 1 

4 

10 

5 

Trial 2 

5 

12 

2 


Is trial 1 or trial 2 in better agreement with the ex- 
pected result? For the moment, however, we leave 
this question aside. 

The probability of getting a particular set of num- 
bers mi in a given series of n trials is found from the 
multinomial expansion of (pi + P 2 + • * • + p v ) n , 
where the pi contain the probability of failure as well 
as of success : 

n 1 

P = — r — , ' Vi m 'VT' ■ ■ ■ P. m - , (32) 

mi\m 2 \ • • • m v \ 


where v = s + 1 and where 2m; = n. It is easily 
shown that (treating m; as a continuous variable) 
this is a maximum for m; = np { . Putting 


P = 

max 


{npi)\{np 2 ) \ • • • (np „) ! 


pfPip^p-i • • • p v 


we have 

_P_ = (np, )I(np»)I • • • 

P max mi \m 2 ! • • • m„! 

This is a product of terms of the form 

m ! 


CONFIDENTIAL 


SAMPLING 


35 



Figure 13. Pearson’s criterion for goodness of fit P (>x 2 )- Contours of probability P(>x 2 ) that fit is good, plotted 
against degrees of freedom s and against squared divergence x 2 - 


Now if m and np are reasonably large, we may put 
( np ) ! = (np/e) np and m ! = ( m/e ) m . Our typical term 
then becomes, after suitable expansions and approxi- 
mations , 5 

— (m—np) 2 /2np 

V > 

a result valid when m and np are not too small. The 
quantity 

v = ( 33 ) 

npi 

is called the divergence of the ith group from its ex- 
pected value. In the following equation we see that 


JL = , 

Pmax 

where 

x 2 = Y] - J2 — o. - n Pi y (34) 

is the total divergence of the series of trials. 

We therefore see that except for the constant 
factor P m ax the probability of getting a given result 
is a function only of % 2 > and rapidly decreases as x 2 
increases. We may use this result to settle the ques- 
tion of which of two results deviates more from the 


CONFIDENTIAL 


36 


PROBABILITY 


expected result: we shall state (by definition) that 
of two given results, the one with the greater value 
of x 2 deviates the more from the expected result. 

The probability of obtaining a result which de- 
viates more than a given result from the expected 
result may now be calculated by direct summation. 
Approximating this sum by an integral leads to the 
answer: 5 

(35> 

where s, the “number of degrees of freedom/’ is equal 
to v — 1. Tables of this function are given in Fry 5 
and other works on statistics. For rough work it may 
be pointed out that P(>x 2 ) is small when % 2 is 
greater than s, and large (near 1) when x 2 is much 
smaller than s. A contour plot of this function is given 
in Figure 13. 

We can now return to Table 6, to point out that 
the trial fits the computed or expected values best 
which gives smallest values of x 2 - In Table 6 we have 
s = 2, npi = 3, np 2 = 12, and npz = 4. In trial 1, 
mi = 4, m 2 = 10, and m 3 = 5. Therefore, 

X 2 = 1 (4-3) 2 + — (10- 12) 2 + - (5— 4) 2 = — , 

3 12 4 12 

whereas, for trial 2, 

X 2 = 1 (5 — 3) 2 + A (12-12) 2 + 1 (2— 4) 2 = 1 . 

Therefore, trial 1 agrees more closely with the ex- 
pected values than does trial 2. In fact, comparing 
these results with Figure 13, we see that the chance 
that the results of trial 1 “really” correspond to the 
expected case (the discrepancy being simply chance 
fluctuation) is 2 in 3; whereas the probability that 
trial 2 corresponds to the expected case is only half 
as great, 1 in 3. 

Usually only the results of a trial are known; we 
have to assume values for the p’s, and compute the 
chance that the true state of things is no farther afield 
than the assumed state. The assumption which gives 
the smallest value of x 2 is the most probable assump- 
tion. 

2.3.2 Some Examples 

In a rocket-firing test, the target consists of two 
concentric rings, one 10 feet in radius, the other 20 


feet in radius. In a trial, 25 rockets are fired. Of these 
10 hit inside the smaller ring, 10 between the rings, 
and 5 outside the rings. We expect from previous 
experience that the hits are distributed according to 
the circular normal law, whose probability density is 

1 — r y 2 <r* 

2x<7 2 

We wish to test the validity of this law, and to de- 
termine a value of a. 

Our method of procedure is to apply the x 2 test to 
these results using various values of a. If P(>x 2 ) is 
always small for all values of a, we have an indication 
that the assumed distribution does not hold. If 
P(>X 2 ) is large for some values of cr, we then know 
that the results are reasonable for those values, and 
the data are not inconsistent with the normal dis- 
tribution. 

It is easily seen that the probability of a shot hit- 
ting inside a ring of radius r is (assuming the proba- 
bility density given above) 

1 - e- rV2 ' 2 . 

Hence the probabilities pi, p 2 , and p 3 of hitting inside 
the inner ring, between the rings, and outside the 
outer ring are 

th = 1 - e~ 50 + 

p 2 = e -* a '°' - e- 200 ^ , 

P 3 = «- 200A \ 

These must be compared with mi = 10, m 2 = 10, 
m 3 = 5. 

Suppose that we begin with the hypothesis that 
a — 10 feet. Then 

Pi = 0.394 , 
p 2 = 0.471 , 

Pz = 0.135 . 

We now proceed as in the following table. 

or = 10 feet 



Actual 

Expected 


Group 

Hits 

Hits 

Difference Divergence 

1 

10 

9.84 

+0.16 0.003 

2 

10 

11.80 

-1.80 0.274 

3 

5 

3.36 

+ 1.64 0.800 

Total 

25 

25.00 

x 2 = 1.077 
P(>x 2 ) =0.58 


Since s = 2, we look up P(> x 2 ) in the table for two 
degrees of freedom. It is seen that a = 10 ft gives a 


CONFIDENTIAL 


SAMPLING 


37 


<r (feet) 

8.6 

8.7 

8.8 


10.8 

10.9 

11.0 


14.2 

14.3 

14.4 

P Ox 2 ) 

0.03 

0.04 

0.07 


0.77 

0.775 

0.76 


0.06 

0.05 

0.04 


reasonable result which means that the normal law 
is reasonable. 

We may proceed in this way and find values of 
P(> X 2 ) for other values of a. The table above shows 
some results. It shows that the most reasonable value 
of <j is 10.9 feet, but that all values between 8.8 feet 
and 14.3 feet are reasonable [P(>x 2 ) > 0.05]. 

Statistical data on antisubmarine flying for three 
months give the following figures : 


Month 

Hours 

flown 

Contacts 

Hours per 
contact 

1 

2,600 

5 

520 

2 

3,500 

6 

583 

3 

4,000 

6 

667 

Total 

10,100 

17 

Avg 595 


The hours per contact seem to be rising, and we wish 
to know if the increase is significant. 

To test this, let us test the hypothesis that the 
hours per contact has remained constant at the aver- 
age value 595. We may then calculate P(>x 2 ) as in 
the following table. 


Expected 

Month Contacts contacts 


Difference Divergence 


1 5 

2 6 

3 6 


4.4 

0.6 

0.08 

6.0 

0.0 

0.00 

6.8 

0.8 

0.09 


x 2 = 0 . 17 

8 = 2 P(>x 2 ) =0.98 

Obviously, the difference may very well be due to 
chance. 


CONFIDENTIAL 


Chapter 3 

THE USE OF MEASURES OF EFFECTIVENESS 


T he material in the preceding chapter, and much 
that is included in the following chapters, is in 
the nature of tools which the operations research 
worker finds useful. A familiarity with these tech- 
niques is necessary for the worker, but it is not in 
itself a guarantee that the worker will be successful 
in operations research. Just as with every other field 
of applied science, the improvement of operations of 
war by the application of scientific analysis requires a 
certain flair which comes with practice, but which is 
difficult to put into words. 

It is important first to obtain an overall quantita- 
tive picture of the operation under study. One must 
first see what is similar in operations of a given kind 
before it will be worthwhile seeing how they differ 
from each other. In order to make a start in so com- 
plex a subject, one must ruthlessly strip away details 
(which can be taken into account later), and arrive 
at a few broad, very approximate “constants of the 
operation.” By studying the variations of these con- 
stants, one can then perhaps begin to see how to im- 
prove the operation. 

It is well to emphasize that these constants which 
measure the operation are useful even though they 
are extremely approximate; it might almost be said 
that they are more valuable because they are very ap- 
proximate. This is because successful application of 
operations research usually results in improvements 
by factors of 3 or 10 or more. Many operations are 
ineffectively compared to their theoretical optimum 
because of a single faulty component: inadequate 
training of crews, or incorrect use of equipment, or 
inadequate equipment. Usually, when the “bottle- 
neck” has been discovered and removed, the im- 
provements in effectiveness are measured in hun- 
dreds or even thousands of per cent. In our first study 
of any operation we are looking for these large factors 
of possible improvement. They can be discovered if 
the constants of the operation are given only to one 
significant figure, and any greater accuracy simply 
adds unessential detail. 

One might term this type of thinking “hemibel 
thinking.” A bel is defined as a unit in a logarithmic 
scale a corresponding to a factor of 10. Consequently, 

a This suggests the advantage of using logarithmic graph 
paper in plotting data. Unity is zero hemibels, 3 is 1 hemibel, 
10 is 2 hemibels, 30 is 3 hemibels, and 10,000 is 8 hemibels. A 
hemibel is 5 decibels. An appropriate abbreviation would be 
hb, corresponding to db for decibel. 


a hemibel corresponds to a factor of the square root 
of 10, or approximately 3. Ordinarily, in the prelimi- 
nary analysis of an operation, it is sufficient to locate 
the value of the constant to within a factor of 3. 
Hemibel thinking is extremely useful in any branch 
of science, and most successful scientists employ it 
habitually. It is particularly useful in operations 
research. 

Having obtained the constants of the operation 
under study in units of hemibels (or to one significant 
figure), we take our next step by comparing these 
constants. We first compare the value of the con- 
stants obtained in actual operations with the opti- 
mum theoretical value, if this can be computed. If 
the actual value is within a hemibel (i.e., within a 
factor of 3) of the theoretical value, then it is ex- 
tremely unlikely that any improvement in the details 
of the operation will result in significant improve- 
ment. In the usual case, however, there is a wide gap 
between the actual and theoretical results. In these 
cases a hint as to the possible means of improvement 
can usually be obtained by a crude sorting of the 
operational data to see whether changes in personnel, 
equipment, or tactics produce a significant change in 
the constants. In many cases a theoretical study of 
the optimum values of the constants will indicate 
possibilities of improvement. 

The present chapter will give a few examples of the 
sort of constants which can be looked for, and the sort 
of conclusions which may be drawn from their study. 

3.1 SWEEP RATES 

An important function for some naval forces, par- 
ticularly for some naval aircraft, is that of scouting 
or patrol, that is, search for the enemy. In submarine 
warfare search is particularly important. The sub- 
marine must find the enemy shipping before it can 
fire its torpedoes, and the antisubmarine craft must 
find the enemy submarine in order to attack it, or to 
route its convoys evasively, and so on. 

Patrol or search is an operation which is peculiarly 
amenable to operations research. The action is sim- 
ple, and repeated often enough under conditions 
sufficiently similar to enable satisfactory data to be 
accumulated. From these data measures of effec- 
tiveness can be computed periodically from which a 
great deal can be deduced. By comparing the opera- 


38 


CONFIDENTIAL 


SWEEP RATES 


39 


tional values of the constants with the theoretically 
optimum values, one can obtain an overall picture 
as to the efficiency of our own forces. Sudden changes 
in the constants without change in our own tactics 
will usually mean a change in enemy tactics which, 
of course, needs investigation and usually counter- 
action. 


3.1.1 Calculation of Constants 

In the simplest case a number of search units (e.g., 
aircraft or submarine) are sent into a certain area A 
of the ocean to search for enemy craft. A total of T 
units of time (hours or days) is spent by one or an- 
other of the search craft in the area, and a number of 
contacts C with an enemy unit are reported. It is 
obvious that the total number of contacts obtained 
in a month is not a significant measure of the effec- 
tiveness of the searching craft because it depends on 
the length of time spent in searching. A more useful 
constant would be the average number of contacts 
made in the area per unit of time spent in searching 
(C divided by T). 

The number of contacts per unit of searching time 
is a simple measure which is useful for some purposes 
and not useful for others. As long as the scene of the 
search remains the same, the quantity ( C/T ) de- 
pends on the efficiency of the individual searching 
craft and also on the number N of enemy craft which 
are in the area on the average. Consequently, any 
sudden change in this quantity would indicate a 
change in enemy concealment tactics, or else a change 
in the number of enemy craft present. Since this 
quantity depends so strongly on the enemy’s actions, 
it is not a satisfactory one to compare against theo- 
retically optimum values in order to see whether the 
searching effort can be appreciably improved or not. 
Nor is it an expedient quantity to use in comparing 
the search efforts in two different areas. 

A large area is more difficult to search over than a 
small one since it takes more time to cover the larger 
area with the same density of search. Consequently, 
the number of contacts per unit searching time should 
be multiplied by the area searched over in order to 
compensate for this area effect, and so that the 
searching effort in two different areas can be com- 
pared on a more or less equal basis. 

312 Operational Sweep Rate 

One further particularly profitable step can be 
taken, if other sources of intelligence allow one to 


estimate (to within a factor of 3) the average number 
of enemy craft in the area while the search was going 
on. 

The quantity which can then be computed is the 
number of contacts per unit search time, multiplied 
by the area searched over and divided by the esti- 
mated number of enemy units in the area. Since the 
dimensions of this quantity are square miles per 
hour, it is usually called the effective, or operational, 
sweep rate. 

Operational sweep rate : 

n square miles . . 

° P \ N T/ hour (or day) 

C = number of contacts; 

A = area searched over in square miles; 

T = total searching time in hours (or days) ; 

N = probable number of enemy craft in area. 

This quantity is a measure of the ability of a single 
search craft to find a single enemy unit under actual 
operational conditions. It equals the effective area 
of ocean swept over by a single search craft in an 
hour (or day) . 

Another way of looking at this constant is taken 
by remembering that (N / A) is the average density of 
target craft, in number per square mile. Since (C/T) 
is the number of contacts produced per hour (or day) 
Qop = (C/T) -r- (N/A) is the number of contacts 
which would be obtained per hour (or day) if the 
density of target craft were one per square mile. 


3.13 Theoretical Sweep Rate 


Sweep rates can be compared from area to area and 
from time to time, since the effects of different size of 
areas and of different numbers of enemy craft are 
already balanced out. Sweep rates can also be com- 
pared with the theoretical optimum for the craft in 
question. In Division 6, Volume 2B it is shown that 
the sweep rate is equal to twice the “effective lateral 
range of detection” of the search craft equipment, 
multiplied by the speed of the search craft. 

Theoretical sweep rate : 


Qth = 2 Rv 


square miles 
hour (or day) 


(2) 


R = effective lateral range of detection in 
miles; 

v = average speed of search craft in miles 
per hour (or day) . 


CONFIDENTIAL 


40 


THE USE OF MEASURES OF EFFECTIVENESS 


A comparison of this sweep rate with the operational 
value will provide us with the criterion for excellence 
which we need. 

The ratio between Q ov> and Q t h is a factor which de- 
pends both on the effectiveness of our side in using 
the search equipment available, and on the effective- 
ness of the enemy in evading detection. For instance, 
if the search craft is a plane equipped with radar, and 
if the radar is in poor operational condition on the 
average, this ratio will be correspondingly dimin- 
ished. Similarly, if the enemy craft is a submarine, 
then a reduction of the average time it spent on the 
surface would reduce the ratio for search planes using 
radar or visual sighting. The ratio also would be re- 
duced if the area were covered by the searching craft 
in a nonuniform manner, and if the enemy craft 
tended to congregate in those regions which were 
searched least. Correspondingly, the ratio (Q op /Qth) 
will be increased (and may even be greater than 
unity) if the enemy craft tend to congregate in one 
region of the area, and if the searching effort is also 
concentrated there. It can be seen that a comparison 
of the two sweep rates constitutes a very powerful 
means of following the fluctuations in efficacy of the 
search operation as the warfare develops. 

31 - 4 Submarine Patrol 

A few examples will show the usefulness of the 
quantities mentioned here. The first example comes 
from data on the sighting of merchant vessels by 
submarines on patrol. Typical figures are given in 
Table 1. All numbers are rounded off to one or two 


Table 1 . Contacts on merchant vessels by submarines. 


Region 

B 

D 

E 

Area, sq miles, A 

80,000 

250,000 

400,000 

Avg No. ships present, N 

20 

20 

25 

Ship flow through area per day, F 

6 

3 

4 

Sub-days in area, T 

800 

250 

700 

Contacts, C 

400 

140 

200 

Sweep rate, Q op 

2,000 

7,000 

4,500 

Fraction of ship flow ^ 

sighted by a sub, — — 

FT 

0.08 

0.2 

0.07 

Sightings per sub per day 

0.5 

0.6 

0.3 


significant figures, since the estimate of the number of 
ships present in the area is uncertain, and there is no 
need of having the accuracy of the other figures any 
larger. The operational sweep rate (computed from 


the data) is also tabulated. Since the ratio of the 
values of Q for regions B and E is less than 1 hemibel, 
the difference in the sweep rates for those regions is 
probably due to the rather wide limits of error of the 
values of N. The difference in sweep rate between 
areas B and D is probably significant however (it 
corresponds to a ratio of more than a hemibel). In- 
vestigation of this difference shows that the antisub- 
marine activity in region B was considerably more 
effective than in D, and, consequently, the subma- 
rines in region B had to spend more time submerged 
and had correspondingly less time to make sightings. 
The obvious suggestion (unless there are other stra- 
tegic reasons to the contrary) is to transfer some of 
the effort from region B to region D, since the yield 
per submarine per day is as good, and since the 
danger to the submarine is considerably less. 

For purposes of comparison, we compute the theo- 
retical sweep rate. A submarine on patrol covers 
about 200 miles a day on the average, and the aver- 
age range of visibility for a merchant vessel is be- 
tween 15 and 20 miles. The theoretical sweep rate, 
therefore, is about 6,000 to 8,000 square miles per 
day. This corresponds remarkably closely with the 
operational sweep rate in regions D and E. The 
close correspondence indicates that the submarines 
are seeing all the shipping they could be expected to 
see (i.e., with detection equipment having a range of 
15 to 20 miles). It also indicates that the enemy has 
not been at all successful in evading the patrolling 
submarines, for such evasion would have shown up as 
a relative diminution in Q op . The reduced value of 
sweep rate in region B has already been explained. 

Therefore, a study of the sweep rate for subma- 
rines against merchant vessels has indicated (for the 
case tabulated) that no important amount of ship- 
ping is missed because of poor training of lookouts or 
of failure of detection equipment. It has also indi- 
cated that one of the three regions is less productive 
than the other two; further investigation has re- 
vealed the reason. The fact that each submarine in 
region D sighted one ship in every five that passed 
through the region is a further indication of the 
extraordinary effectiveness of the submarines pa- 
trolling these areas. 

3.1.5 Aircraft Search for Submarines 

Another example, not quite so impressive, but per- 
haps more instructive, can be taken from data on 
search for submarines by antisubmarine aircraft. 
Typical values are shown in Table 2, for three suc- 


CONFIDENTIAL 


SWEEP RATES 


41 


cessive months, for three contiguous areas. Here the 
quantity T represents the total time spent by aircraft 
over the ocean on antisubmarine patrol of all sorts 
in the region during the month in question. The 
quantity C represents the total number of verified 
sightings of a surfaced submarine in the area and 
during the month in question. From these data the 
value of the operational sweep rate, Q op , can be com- 
puted and is expressed also on a hemibel scale. From 
these figures a number of interesting conclusions can 
be drawn, and a number of useful suggestions can be 
made for the improving of the operational results. 


rines. During the latter month the submarines car- 
ried on an all-out attack, coming closer to shore than 
before or since, and staying longer on the surface, in 
order to sight more shipping. This bolder policy ex- 
posed the submarines to too many attacks, so they 
returned to more cautious tactics in June. The epi- 
sode serves to indicate that at least one-half of the 
2 hemibel discrepancy between operational and theo- 
retically maximum sweep rates is probably due to the 
submergence tactics of the submarine. 

The other factor of 3 is partially attributable to a 
deficiency in operational training and practice in 


Table 2. Sightings of submarines by antisubmarine aircraft. 


Region 

A 

B 

C 

Area, sq miles, A 

300,000 

600,000 

900,000 

Month 

A 

M 

J 

A 

M 

J 

A 

M 

J 

Avg No. subs, N 

7 

7 

6 

1 

4 

3 

3 

7 

5 

Total plane time (in thousands of hours), T 

20 

25 

24 

6 

7 

9 

5 

5 

6 

Contacts, C 

39 

37 

30 

2 

35 

14 

4 

11 

9 

Sweep rate, Q op 

80 

60 

60 

200 

750 

300 

240 

280 

270 

Sweep rate in hemibels 

4 

4 

4 

5 

6 

5 

5 

5 

5 


We first compare the operational sweep rate with 
the theoretically optimum rate. The usual antisub- 
marine patrol plane flies at a speed of about 150 
knots. The average range of visibility of a surfaced 
U-boat in flyable weather is about 10 miles. There- 
fore, if the submarines were on the surface all of the 
time during which the planes were searching, we 
should expect the theoretical search rate to be 3,000 
square miles per hour, according to equation (2) . On 
the hemibel scale this is a value of 7. If the subma- 
rines on the average spent a certain fraction of the 
time submerged, then Qth would be proportionally 
diminished. We see that the average value of the 
sweep rate in regions B and C is about one-tenth 
(2 hemibels) smaller than the maximum theoretical 
value of 3,000. 

Part of this discrepancy is undoubtedly due to the 
submergence tactics of the submarines. In fact, the 
sudden rise in the sweep rate in region B from April 
to May was later discovered to be almost entirely 
due to a change in tactics on the part of the subma- 


antisubmarine lookout keeping. Antisubmarine pa- 
trol is a monotonous duty. The average plane can 
fly for hundreds of hours (representing an elapsed 
time of six months or more) before a sighting is made. 
Experience has shown that, unless special competitive 
practice exercises are used continuously, performance 
of such tasks can easily fall below one-third of their 
maximum effectiveness. Data in similar circum- 
stances, mentioned later in this chapter, show that a 
diversion of 10 per cent of the operational effort into 
carefully planned practice can increase the overall 
effectiveness by factors of two to four. 

We have thus partially explained the discrepancy 
between the operational sweep rate in regions B and 
C and the theoretically optimum sweep rate; we have 
seen the reason for the sudden increase for one month 
in region B. We must now investigate the result of 
region A which displays a consistently low score in 
spite of (or perhaps because of) the large number of 
antisubmarine flying hours in the region. Search in 
region A is consistently 1 hemibel worse (a factor of 


CONFIDENTIAL 


42 


THE USE OF MEASURES OF EFFECTIVENESS 


3) than in the other two regions. Study of the details 
of the attacks indicates that the submarines were not 
more wary in this region; the factor of 3 could thus 
not be explained by assuming that the submarines 
spent one-third as much time on the surface in region 
A. Nor could training entirely account for the differ- 
ence. A number of new squadrons were “broken in” 
in region A, but even the more experienced squadrons 
turned in the lower average. 

3.1.6 Distribution of Flying Effort 

In this case the actual track plans of the antisub- 
marine patrols in region A were studied in order to 
see whether the patrol perhaps concentrated the 


effort in region A. Flying in the inner zone, where 
three-quarters of the flying was done, is only one- 
tenth as effective as flying in the outer zone, where 
less than 1 per cent of the flying was done. Due per- 
haps to the large amount of flying in the inner zone, 
the submarines did not come this close to shore very 
often, and, when they came, kept well submerged. 
In the outer zones, however, they appeared to have 
been as unwary as in region B in the month of May. 

If a redistribution of flying effort would not have 
changed submarine tactics, then a shift of 2,000 
hours of flying per month from the inner zone to the 
outer (which would have made practically no change 
in the density of flying in the inner zone, but which 
would have increased the density of flying in the outer 


Table 3. Sightings of submarines by antisubmarine planes, offshore effect. 


Distance from shore in miles 

0 to 60 

60 to 120 

120 to 180 

180 to 240 

Flying time in sub-region, T (in thousands of hours) 

15.50 

3.70 

0.60 

0.17 

Contacts made in sub-region, C 

21 

11 

5 

2 

Contacts per 1,000 hours flown, (C/7 7 ) 

1.3 

3 

8 

12 

Contacts per 1,000 hours flown, in hemibels 

0 

1 

2 

2 


flying effort in regions where the submarines were not 
likely to be. This indeed proved to be the case, for it 
was found that a disproportionately large fraction of 
the total antisubmarine flying in region A was too 
close to shore to have a very large chance of finding a 
submarine on the surface. The data for the month of 
April (and also for other months) was broken down 
according to the amount of patrol time spent a given 
distance off shore. The results for the one month are 
given in Table 3. In this analysis it was not necessary 
to compute the sweep rate, but only to compare the 
number of contacts per thousand hours flown in var- 
ious strips at different distances from the shore. This 
simplification is possible since different strips of the 
same region are being compared for the same periods 
of time; consequently, the areas are equal and the 
average distribution of submarines is the same. The 
simplification is desirable since it is not known, even 
approximately, where the seven submarines, which 
were present in that region in that month, were dis- 
tributed among the offshore zones. 

A comparison of the different values of contacts 
per 1,000 hours flown for the different offshore bands, 
immediately explains the ineffectiveness of the search 


zone by a factor of 13) would have approximately 
doubled the number of contacts made in the whole 
region during that month. Actually, of course, when a 
more uniform distribution of flying effort was in- 
augurated in this region, the submarines in the outer 
zones soon became more wary and the number of 
contacts per thousand hours flown in the outer region 
soon dropped to about 4 or 5. This still represented a 
factor of 3, however, over the inshore flying yield. 
We therefore can conclude that the discrepancy of 
one hemibel in sweep rate between region A and 
regions B and C is primarily due to a maldistribution 
of patrol flying in region A, the great preponderance 
of flying in that region being in localities where the 
submarines were not. When these facts were pointed 
out, a certain amount of redistribution of flying was 
made (within the limitations imposed by other fac- 
tors), and a certain amount of improvement was 
observed. 

The case described here is not a unique one; in fact, 
it is a good illustration of a situation often encoun- 
tered in operations research. The planning officials 
did not have the time to make the detailed analysis 
necessary for the filling in of Table 3. They saw that 


CONFIDENTIAL 


SWEEP RATES 


43 


many more contacts were being made on submarines 
close inshore than farther out, and they did not have 
at hand the data to show that this was entirely due to 
the fact that nearly all the flying was close to shore. 
The data on contacts, which is more conspicuous, 
might have actually persuaded the operations officer 
to increase still further the proportion of flying close 
to shore. Only a detailed analysis of the amount of 
flying time in each zone, resulting in a tabulation of 
the sort given in Table 3, was able to give the officer 
a true picture of the situation. When this had been 
done, it was possible for the officer to balance the 
discernible gains to be obtained by increasing the 
offshore flying against other possible detriments. In 
this case, as with most others encountered in this 
field, other factors enter; the usefulness of the patrol 
planes could not be measured solely by their collec- 
tion of contacts, and the other factors favored inshore 
flying. 

3.1.7 Antisubmarine Flying in the 

Bay of Biscay 

An example of the use of sweep rate for following 
tactical changes in a phase of warfare will be taken 
from the RAF Coastal Command struggle against 
German U-boats in the Bay of Biscay. After the 
Germans had captured France, the Bay of Biscay 
ports were the principal operational bases for U- 
boats. Nearly all of the German submarines operat- 
ing in the Atlantic went out and came back through 
the Bay of Biscay. About the beginning of 1942, 
when the RAF began to have enough long range 
planes, a number of them were assigned to antisub- 
marine duty in the Bay to harass these transit U- 
boats. Since the submarines had to be discovered 
before they could be attacked, and since these planes 
were out only to attack submarines, a measure of the 
success of the campaign was the number of U-boat 
sightings made by the aircraft. 

The relevant data for this part of the operation are 
shown in Figure 1 for the years 1942 and 1943. The 
number of hours of antisubmarine patrol flying in 
the Bay per month, the number of sightings of 
U-boats resulting, and the estimated average num- 
ber of U-boats in the Bay area during the month are 
plotted in the upper part of the figure. From these 
values and from the area of the Bay searched over 
(130,000 square miles), one can compute the values 
of the operational sweep rate which are shown in the 
lower half of the figure. 


The graph for Q op indicates that two complete 
cycles of events have occurred during the two years 
shown. The first half of 1942 and the first half of 1943 
gave sweep rates of the order of 300 square miles per 
hour, which correspond favorably with the sweep 
rates obtained in regions B and C in Table 2. The 
factor of 10 difference between these values and the 
theoretically maximum value of 3,000 square miles 
per hour can be explained, as before, partly by the 
known discrepancy between lookout practice in 
actual operation and theoretically optimum lookout 
effectiveness, and mainly by submarine evasive tac- 
tics. It was known at the beginning of 1942 that the 
submarines came to the surface for the most part at 
night, and stayed submerged during a good part of 
the day. Since most of the antisubmarine patrols 
were during daylight, these tactics could account for 
a possible factor of 5, leaving a factor of 2 to be 
accounted for (perhaps) by lookout fatigue, etc. 

During the early part of 1942, the air cover over 
the Bay of Biscay increased, and the transit sub- 
marines began to experience a serious number of 
attacks. In the spring a few squadrons of radar planes 
were equipped for night-flying, with searchlights to 
enable them to make attacks at night on the sub- 
marines. When these went into operation, the effec- 
tive search rate for all types of planes increased at 
first. The night-flying planes caught a large number 
of submarines on the surface at night. These night 
attacks caused the submarines to submerge more at 
night and surface more in the daytime; therefore the 
day-flying planes also found more submarines on the 
surface. 

The consequent additional hazard to the U-boats 
forced a countermeasure from the Germans; for even 
though the night-flying was a small percentage of the 
total air effort in the Bay, the effects of night attack 
on morale were quite serious. The Germans started 
equipping their submarines with radar receivers cap- 
able of hearing the L-band radar set carried in the 
British planes. When these sets were operating prop- 
erly, they would give the submarine adequate warn- 
ing of the approach of a radar plane, so that it could 
submerge before the plane could make a sighting or 
attack. Despite difficulties in getting the sets to work 
effectively, they became more and more successful, 
and the operational sweep rate for the British planes 
dropped abruptly in the late summer of 1942, reach- 
ing a value about one-fifth of that previously at- 
tained. 

When this low value of sweep rate continued for 


CONFIDENTIAL 


SIGHTINGS OF U-BOATS PER MONTH 


44 


THE USE OF MEASURES OF EFFECTIVENESS 


1942 


1943 


J FMAMJJ ASONDJ FMAMJJ ASONDJ 



cn 


< 

o 


10,000 

6000 

4000 

2000 


— 1000 


600 

400 


200 


cn 

f- 

< 

o 

CD 



<0 


Ui 

CO 


UI 

X 


< 

cr 

CL 

UJ 

UI 

£ 

</> 


or 

ui 

Q- 

-I 

O 

or 

H* 

2 


co 

cc 

X 

o 

X 

o 


CO 

rfi 

05 

1—1 

I 

<N 

05 


o3 


PQ 


PQ 

05 


o 

c 3 

g 

-S 

o 


0 

cS 

>> 


o3 

O 

X 2 


bfl 

g 

"-4J 

# bC 

m 


H 

£> 

O 

£ 


1942 


1943 - 


C0NFIDENTIAL 


EXCHANGE RATES 


45 


several months, it was obviously necessary for the 
British to introduce a new measure. This was done 
by fitting the antisubmarine aircraft with S-band 
radar which could not be detected by the L-band re- 
ceivers on the German submarines at that time. 
Commencing with the first of 1943, the sweep rate 
accordingly rose again as more and more planes were 
fitted with the shorter wave radar sets. Again the 
U-boats proved particularly susceptible to the at- 
tacks of night-flying planes equipped with the new 
radar sets and with searchlights. By midsummer of 
1943, the sweep rate was back as high as it had been 
a year before. 

The obvious German countermeasure was to equip 
the submarines with S-band receivers. This, however, 
involved a great many design and manufacturing 
difficulties, and these receivers were not to be avail- 
able until the fall of 1943. In the interim the Ger- 
mans sharply reduced the number of submarines sent 
out, and instructed those which did go out to stay 
submerged as much as possible in the Bay region. 
This reduced the operational sweep rate for the RAF 
planes to some extent, and, by the time the U-boats 
had been equipped with S-band receivers in the fall, 
the sweep rate reached the same low values it had 
reached in the previous fall. The later cycle, which 
occurred in 1944, involved other factors which we 
will not have time to discuss here. 

This last example shows how it is sometimes pos- 
sible to watch the overall course of a part of warfare 
by watching the fluctuations of a measure of effec- 
tiveness. One can at the same time see the actual 
benefits accruing from a new measure and also see 
how effective are the countermeasures. By keeping a 
month-to-month chart of the quantity, one can time 
the introduction of new measures, and also can assess 
the danger of an enemy measure. A number of other 
examples of this sort will be given later in this chapter. 

3 2 EXCHANGE RATES 

A useful measure of effectiveness for all forms of 
warfare is the exchange rate, the ratio between enemy 
loss and own loss. Knowledge of its value enables one 
to estimate the cost of any given operation and to 
balance this cost against other benefits accruing from 
the operation. Here again a great deal of insight can 
be obtained into the tactical trends by comparing 
exchange rates; in particular, by determining how 
the rate depends on the relative strength of the forces 
involved. 


When the engagement is between similar units, as 
in a battle between tanks or between fighter planes, 
the units of strength on each side are the same, and 
the problem is fairly straightforward. Data are 
needed on a large number of engagements involving a 
range of sizes of forces involved. Data on the strength 
of the opposed forces at the beginning of each engage- 
ment and on the resulting losses to both sides are 
needed. These can then be subjected to statistical 
analysis to determine the dependence of the losses on 
the other factors involved. 

Suppose m and n are the number of own and enemy 
units involved, and suppose k and l are the respective 
losses in the single engagements. In general, k and l 
will depend on m and n, and the nature of the depen- 
dence is determined by the tactics involved in the 
engagement. For instance, if the engagement con- 
sists of a sequence of individual combats between 
single opposed units, then both k and l are propor- 
tional to either m or n (whichever is smaller), and the 
exchange rate (l/k) is independent of the size of the 
opposing forces. On the other hand, if each unit on 
one side gets about an equal chance to shoot at each 
unit on the other side, then the losses on one side will 
be proportional to the number of opposing units 
(that is, k will be proportional to n, and l will be pro- 
portional to m). These matters will be discussed in 
further detail and from a somewhat different point 
of view in the next chapter. 

3 . 2.1 Air-to-Air Combat 

The engagements between American and Japanese 
fighter aircraft in the Pacific in 1943-44 seem to have 
corresponded more closely to the individual-combat 
type of engagements. The data which have been 
analyzed indicate that the exchange rate for Japanese 
against U. S. fighters (l/k) was approximately inde- 
pendent of the size of the forces in the engagement. 
The percentage of Japanese fighters lost per engage- 
ment seems to have been independent of the numbers 
involved (i.e., k was proportional to n); whereas the 
percentage of U. S. fighters lost per engagement 
seemed to increase with an increase of Japanese 
fighters, and decrease with an increase of U. S. 
fighters (i.e., I was also proportional to n). 

The exchange rate for U. S. fighters in the Pacific 
during the years 1943 and 1944 remained at the sur- 
prisingly high value of approximately 10. This cir- 
cumstance contributed to a very high degree to the 
success of the U. S. Navy in the Pacific. It was, there- 


CONFIDENTIAL 


46 


THE USE OF MEASURES OF EFFECTIVENESS 


fore, of importance to analyze as far as possible the 
reasons for this high exchange rate in order to see the 
importance of the various contributing factors, such 
as training and combat experience, the effect of the 
characteristics of planes, etc. The problem is natur- 
ally very complex, and it is possible here only to give 
an indication of the relative importance of the con- 
tributing factors. 

Certainly a very considerable factor has been the 
longer training which the U. S. pilots underwent com- 
pared to the Japanese pilots. A thoroughgoing study 
of the results of training and of the proper balance 
between primary training and operational practice 
training has not yet been made, so that a quantitative 
appraisal of the effects of training is as yet impossible. 
Later in this chapter we shall give an example which 
indicates that it sometimes is worth while even to 
withdraw aircraft from operations for a short time in 
order to give the pilots increased training. There is 
considerable need for further operational research in 
such problems. It is suspected that, in general, 
the total effectiveness of many forces would be 
increased if somewhat more time were given to re- 
fresher training in the field, and slightly less to opera- 
tions. 

The combat experience of the pilot involved has 
also had its part in the high exchange rate. The RAF 
Fighter Command Operations Research Group has 
studied the chance of a pilot being shot down as a 
function of the number of combats the pilot has been 
in. This chance decreases by about a factor of 3 from 
the first to the sixth combat. A study made by the 
Operations Research Group, U. S. Army Air Forces, 
indicates that the chance of shooting down the enemy 
when once in a combat increases by 50 per cent or 
more with increasing experience. 

The exchange rate will also depend on the types of 
planes entering the engagement. An analysis of 
British-German engagements indicates that Spitfire 
9 has an exchange rate about twice that of Spitfire 5. 
The difference is probably mostly due to the differ- 
ence in speed, about 40 knots. There are indications 
that the exchange rate for F6F-5 is considerably 
larger than that for the F6F-3. Since the factors of 
training, experience, and plane type all appear to 
have been in the favor of the United States, it is not 
surprising that the exchange rate turned out to be as 
large as 10. Nevertheless, it would be of interest to 
carry out further analysis to determine which of 
these factors is the most important. 


3.2.2 Convoys versus Submarines 

When the engagement is between units of different 
sorts, the problem becomes more complicated. For 
one thing, a complete balance of gain and loss can 
only be obtained when it is possible to compare the 
value of one unit with one of a different type. The 
question of the relative values of different units in an 
engagement will be taken up later in this chapter. In 
some cases of mixed engagements, however, it is 
possible to gain a considerable insight into the dy- 
namics of the warfare without having to go into the 
vexing question of comparative values. 

An interesting example of a mixed engagement is 
the attack on convoys by submarines. Here an addi- 
tional factor enters the picture, the number of escort 
vessels. Therefore there are three forces entering 
each engagement, the number of merchant vessels in 
the convoy, m, the number of escort vessels, c, and 
the number of U-boats in the attacking pack, n. The 
two losses during the engagement which are of inter- 
est here are k, the number of merchant vessels sunk 
per pack attack, and l, the number of U-boats sunk 
per engagement. The exchange rate of interest here 
is (l/k), the number of submarines sunk per merchant 
vessel sunk. 

As an example we will consider the data on the 
attacks on North Atlantic convoys during the years 
1941 and 1942. The time is chosen after the Germans 
had introduced their wolf-pack tactics, and before 
the introduction of the escort carriers, so that the 

Table 4. U-boat attacks on convoys in North Atlan- 
tic, 1941-1942. m = no. M/V in convoy; c = no. 
escorts; n = no. U/B in pack; k = M/V sunk per en- 
gagement; l = U/B sunk per engagement. 

Independence of M /V sunk on convoy size. 


m ( ran ^ e 
\ mean 

15-24 

20 

25-34 

30 

35-44 

39 

45-54 

48 

No. engagements 

8 

11 

13 

7 

k , mean 

5 

6 

6 

5 

c, mean 

7 

7 

6 

7 

n, mean 

7 

5 

6 

5 


period was one of comparative stability. The first, 
and perhaps the most important aspect of the data, 
is that the number of merchant vessels sunk per pack 
attack turned out to be independent of the number of 
merchant vessels in convoy. This is shown in Table 4. 
Here the data are sorted out according to size of 


CONFIDENTIAL 


EXCHANGE RATES 


47 


convoy, and spreads over a range of nearly 1 hemibel. 
Nevertheless the mean value of A; for each value of m 
is independent of m within the accuracy of the data. 
The mean values of c and n are also given for the data 
chosen, to show that their averages are fairly con- 
stant, and therefore that the results are not due to a 
counterbalancing trend in these quantities. As far as 
the data show, no more vessels are sunk on the aver- 
age from a large convoy than are sunk from a small 
convoy when attacked. In other words, the percentage 
of vessels sunk from a large convoy is smaller than the 
percentage of vessels sunk from a small convoy. This 
is the fundamental fact which makes convoying 
profitable. 

The number of merchant vessels sunk per engage- 
ment does depend upon the number of escort vessels 
and on the number of U-boats in the pack, however. 
The dependence of k on U-boat pack size is shown in 
Table 5. Here the data are sorted out according to n 


Table 5. Dependence of M/V sunk on U/B pack size. 


/ ran s e 

\ mean 

1 

1 

2-5 

3.6 

6-9 

7 

10-15 

14 

Averages 

(weighted) 

No. engage- 
ments 

29 

32 

22 

5 

88 total 

k, mean 

0.9 

3 

4 

6 


c, mean 

6 

7 

7 

8 

6.7 

C kc/n ) 

5.4 

5.8 

4.0 

3.4 

5.1 


over a range of more than two hemibels. The quan- 
tity k itself changes by a factor of two hemibels over 
this range, but the quantity ( k/n ) stays constant 
within the accuracy of the data. 

The dependence of the quantity k on the number 
of escort vessels in the convoy is shown in Table 6. 


Table 6. Dependence of M/V sunk on no. escorts. 


/ range 
\ mean 

1-3 

2 

4-6 

5 

7-9 

8 

10-12 

11 

13-15 

14 

Averages 

(weighted) 

No. engage- 
ments 

6 

42 

25 

13 

2 

88 total 

k, mean 

4.5 

3.4 

3.0 

1.1 

2.0 


n, mean 

3 

4 

4 

2 

10 

3.8 

(kc/n) 

3.0 

4.2 

6.0 

6.0 

2.8 

4.9 


Here the data are sorted out according to c over a 
range of nearly 2 hemibels. Unfortunately a sorting 
according to c has also meant a partial sorting accord- 
ing to n, so that the value of k fluctuates rather 
widely. It is perhaps allowable to say that k is in- 
versely proportional to c, although this inverse de- 


pendence does not seem to hold for c as small as 
unity. We are here looking for major changes, how- 
ever, and fine points should first be set aside. 

Consequently we can say that, within the accuracy 
which we are considering here, and over the inter- 
mediate range of values of escort size and U-boat 
size, the number of merchant vessels lost per pack 
attack is proportional to the number of U-boats in 
the pack, and is roughly inversely proportional to 
the number of escorts. 


3.2.3 Resulting Exchange Rates 

A similar analysis of the submarines sunk during 
these pack attacks shows that l, the number of U- 
boats sunk per attack, is proportional to the number 
of U-boats in the pack, n, and also proportional to 
the number of escorts protecting the convoy, c. To 
the approximation considered here, then, the two 
quantities turn out to be dependent on the forces 
involved in the manner shown in equation (3). The 
corresponding exchange rate is also given in this 
equation : 



As pointed out before, the dependence of k and l on 
n and c does not extend to the limits of very small or 
very large values. Nevertheless the equations seem 
to be reasonably valid in the ranges of practical 
interest. 

The important facts to be deduced from this set of 
equations seem to be: (1) the number of ships lost 
per attack is independent of the size of the convoy, 
and (2) the exchange rate seems to be proportional 
to the square of the number of escort vessels per 
convoy. This squared effect comes about due to the 
fact that the number of merchant vessels lost is 
reduced, and at the same time the number of U-boats 
lost per attack is increased, when the escorts are in- 
creased, the effect coming in twice in the exchange 
rate. The effect of pack size cancels out in the ex- 
change rate. From any point of view, therefore, the 
case for large convoys is a persuasive one. 

When the figures quoted here were presented to 
the appropriate authorities, action was taken to in- 
crease the average size of convoys, thereby also in- 


CONFIDENTIAL 


48 


THE USE OF MEASURES OF EFFECTIVENESS 


creasing the average number of escort vessels per 
convoy. As often occurs in cases of this sort, the 
eventual gain was much greater than that predicted 
by the above reasoning, because by increasing convoy 
and escort size the exchange rate (U/B sunk)/ (M/V 
sunk) was increased to a point where it became un- 
profitable for the Germans to attack North Atlantic 
convoys, and the U-boats went elsewhere. This de- 
feat in the North Atlantic contributed to the turning 
point in the “Battle of the Atlantic.” 

3.3 COMPARATIVE EFFECTIVENESS 

In many cases of importance it is necessary to 
compare the relative effectiveness of two different 
weapons or tactics in gaining some strategic end. It is 
possible to destroy enemy shipping, for instance, by 
using submarines or by using aircraft; it is possible to 
combat enemy submarines by attacking them on the 
high seas or while they are in harbor, refueling; it is 
possible to use aircraft in attacking enemy front-line 
troops or in destroying munitions factories. Such 
comparisons are always difficult. It is often hard to 
find a common unit of measure, and frequently po- 
litical and other nonquantitative aspects must enter 
into the decision. Nevertheless, in these cases it is 
important and useful that the operations research 
worker be able to make as objective and quantitative 
a comparison as possible in order to insure that emo- 
tional and personal arguments do not carry the de- 
cision by default. 

In such cases an important part of the problem lies 
in the choice of an equitable and usable unit of com- 
parison. Care must be taken lest the choice of units 
prejudice the results by omitting important aspects 
of the problem. In fact, it is sometimes almost im- 
possible to find a practicable unit of measure which 
does not prejudge the problem to some extent. It is 
therefore important for the operations research 
worker to estimate as objectively as possible what 
aspects of the problem must be measured and what 
can be neglected without vitiating the results. Some 
important imponderables must be left out because 
they cannot be expressed in quantitative terms. 
These omissions must be recognized, so that they 
may be given their proper weight in the final deci- 
sion. For instance, the effects of bombing or of area 
gunfire on morale are matters which cannot be ade- 
quately expressed in numbers. It is best, therefore, 
when discussing the effects of bombing or area fire, 
to confine the numbers to physical results and to 


point out that the resulting numbers do not include 
the effect on morale. 

Where disparate tactics are to be compared, it is 
not to be expected that the quantitative comparison 
will be at all accurate. Hence, unless the results for 
the alternative methods differ by factors of a hemibel 
or more, one should conclude that the alternatives 
are effectively equivalent. One might say that the 
nonquantitative aspects of the decision would often 
be able to counterbalance differences of factors of two 
or less, but should not outweigh order-of-magnitude 
differences. 

3.3.1 Effectiveness of Anti-Ship Weapons 

In many naval problems it is important to be able 
to assess the relative importance of ship damage to 
ship sinking. In such cases a profitable measure of 
comparison is the amount of time a dockyard will 
take to make up a loss. A damaged ship requires so 
much dockyard time for repair, and a ship sunk re- 
quires so much time to build a replacement. Until 
this time is made up, the ship will not be back in 
service, and no amount of money or trained personnel 
can provide its equivalent meanwhile. A comparison 
of the various methods of attacking shipping can 
therefore be given in terms of the number of ship- 
months lost by the enemy, and a comparison of dif- 
ferent defensive methods can be given in terms of the 
number of shipmonths gained by our side. 

An interesting example of this type of comparison 
is given by a study by the Director of Naval Opera- 
tions Research, Admiralty, on the relative impor- 
tance of different types of protective armor on British 
cruisers. In World War II England had a number of 
her cruisers damaged or lost by various causes : shells 
from enemy naval vessels, bombs, mines, and tor- 
pedoes. The purpose of the study was to assess the 
relative importance of the damage due to these four 
causes and thereby to show what it was most impor- 
tant to defend against. 

In this study the effects of the damage were meas- 
ured by giving the number of months the cruiser was 
out of service for repairs. The equivalent value of a 
cruiser sunk was taken to be 36 cruiser-months since 
it takes about this time to build a new cruiser. In 
addition to giving an indication of the cost of repair- 
ing or replacing the casualties, the cruiser-month loss 
measure reflects the degree to which the Navy was 
immobilized as a result of the attack. 

The data in Table 7 show a number of interesting 


CONFIDENTIAL 


COMPARATIVE EFFECTIVENESS 


49 


Table 7. Casualties to cruisers by enemy action. 


Cause 

Shell 

Bomb 

Mine 

Tor- 

pedo 

Total 

Ships sunk 

3 

9 

1 

11 

24 

Ships damaged 

18 

56 

9 

19 

102 

Total casualties 

21 

65 

10 

30 

126 

Cruiser- ( b y sink ing 

no 

320 

40 

400 

870 

months b y damage 

30 

90 

60 

180 

360 

lost Total 

140 

410 

100 

580 

1,230 

[ Per cent 

11 

34 

8 

47 

100 

Cruiser-months per 






casualty 

7 

6 

10 

19 

10 


points. In the first place, the number of cruiser 
casualties (sunk and damaged) as a result of bombing 
attacks were more than 50 per cent of all casualties, 
but the number of cruiser-months lost per casualty 
due to bombing attacks was less than for the other 
types of attack. In fact in terms of cruiser-months 
lost, torpedo attacks were considerably more impor- 
tant; a torpedo casualty turns out to be about three 
times as serious as a bomb casualty. A further study 
of the bombing casualties indicated that most of the 
cruisers sunk by this means, corresponding to more 
than half of the cruiser-months lost by bombing, 
were sunk by the effect of underwater damage caused 
by near misses. Consequently, a great deal more than 
half of the total cruiser-months lost due to enemy 
action has come from underwater damage to the 
ship’s structure. Most of the rest of the cruiser- 
months lost due to bombing were the result of fires 
started by direct hits of bombs. 

The conclusions indicated from this table are not 
difficult to reach : more attention should be given to 
fire control equipment and training, and new cruisers 
should be designed with better underwater protec- 
tion, even if it means the sacrifice of some above- 
water armor. 

3.3.2 Bombing U-Boat Pens versus Escorting 
Convoys 

A similar, though more complicated, analysis was 
used in studying the question of the relative value of 
using aircraft to escort convoys, to bomb submarine 
base facilities, or to hunt down submarines in the Bay 
of Biscay. In this case the unit of effort is the sortie, 
an individual flight by a plane. The unit of gain is, 


of course, the reduction in the number of ships sunk. 
The time chosen for the example is the last six months 
of 1942, and the place is the waters within aerial 
range of Great Britain. 

During this time convoys from England were 
attacked or threatened with attack about one-tenth 
of the time. Some of the threatened convoys were 
given air escort protection, and others were not. 
Those not protected had a higher loss rate than those 
which were protected by aircraft, so that the aircraft 
actually saved ships from being sunk. The data for 
the last six months of 1942 for Coastal Command 
aircraft show that every hundred sorties flown to 
protect threatened convoys saved about 30 ships 
from being sunk. Consequently, if the aircraft were 
used only to protect threatened convoys, their anti- 
submarine (or “shipping-protective”) efficiency was 
extremely high, for they saved about 30 ships per 
hundred sorties flown. If, however, all convoys have 
to be protected all the time in order to insure protec- 
tion when the convoy is threatened, then the plane’s 
effectiveness is diluted, and only about 3 ships are 
saved per hundred sorties of ordinary escort flying. 

Turning now to the use of aircraft in bombing the 
U-boat repair and refitting bases in the Bay of Bis- 
cay, we must rely on the assessments of damage ob- 
tained from photographic reconnaissance after each 
bombing raid. No submarines were sunk in port, but 
there was enough damage to the bases to slow down 
the refitting of the submarines and therefore to keep 
them off the high seas. It was estimated that about 
15 U-boat months were lost due to the damaging 
effect of the raids. At that time each submarine on 
the average sank about 0.8 ship per month on patrol. 
Consequently a loss of 15 U-boat months due to 
damage of repair facilities represented a gain to our 
side of about 12 ships which were not sunk. This was 
accomplished by a series of raids which totaled about 
1,100 sorties. Consequently the gain to our side was 
about one ship saved per hundred sorties of effort 
against the U-boat bases. This effort is not as effec- 
tive as escorting convoy and is far less effective than 
protecting convoys which are threatened with attack. 

The use of aircraft for antisubmarine patrol in the 
Bay of Biscay is an example of the use of offensive 
tactics for a defensive strategical task. The imme- 
diate result of the patrols is a number of submarines 
sunk plus a number more delayed in passage through 
the Bay. The final result, however, is in saving our 
ships from being sunk. The average life of a subma- 
rine on patrol at that time was about ten months. 


CONFIDENTIAL 


50 


THE USE OF MEASURES OF EFFECTIVENESS 


If it were on patrol in the North Atlantic convoy re- 
gion at that time, it sank about eight ships before it 
was sunk. From this point of view, therefore, each 
submarine sunk in the Bay of Biscay represented a 
net saving of about eight ships. From another point 
of view, however, it is perhaps better to estimate the 
equivalent amount of time lost by the Germans in 
replacing the sunk submarines. In 1942 there were 
enough submarines being constructed so that the 
bottleneck was in the training of the crews. This 
training period (done in the Baltic) took about six 
months, so that one could say that a submarine 
sunk was equivalent to about six submarine-months 
lost. Therefore, from this point of view, a submarine 
sunk was equivalent to about six ships saved, which 
corresponds fairly well with the other estimate of 
eight ships saved. 

In the last half of 1942, aircraft patrol in the Bay 
of Biscay sighted about six U-boats per hundred 
sorties and sank about one-half a U-boat per hundred 
sorties. In addition, it is estimated that a hundred 
sorties produced a net delaying effect on the subma- 
rines in transit of about one U-boat month, corre- 
sponding to one ship saved. Therefore, the net effect 
of offensive patrol in the Bay of Biscay area corre- 
sponded to between four and five ships saved per 
hundred sorties, an effectiveness which is somewhat 
larger than continuous escort of convoy, is consider- 
ably larger than the effectiveness of bombing the 
U-boat bases, but which is considerably smaller than 
the effectiveness of protecting threatened convoys. 
Naturally, these relative effectiveness values changed 
as the war progressed. 

The above comparative figures are not the whole 
basis upon which a decision as to employment of air- 
craft should be reached. Nevertheless, they are a part 
of the material which must be considered, and the de- 
cision would be less likely to be correct if these figures 
were not available. Presumably, if there were a very 
small amount of flying effort which could be ex- 
pended in antisubmarine work, then the planes 
should be assigned to the protection of threatened 
convoys. With a somewhat greater number of planes 
available, it probably would be advisable to spend 
part of that effort in the Bay of Biscay. Another 
consideration also enters into the problem: the fact 
that it might be possible to divert bombers from 
other missions to bomb the U-boat bases from time 
to time, whereas it would not be possible to use these 
same bombers to escort convoys or patrol the Bay 
of Biscay. 


3 . 3.3 Submarine versus Aircraft as Anti-Ship 
Weapons 

A comparison between submarines and aircraft in 
sinking enemy shipping is another question of con- 
siderable interest, but one which raises still more non- 
quantitative considerations. We can attempt to com- 
pute, however, the expected number of enemy ships 
sunk by the average operational submarine and com- 
pare it with the number of ships sunk by an aircraft 
used as efficiently as possible in the same region. The 
problem in both cases divides itself into two ques- 
tions: the number of ships sighted per operational 
month, and the average number of ships sunk per 
sighting. The first question involves the values of 
sweep rate which have been discussed in Section 3.1. 
For instance, the average commissioned submarine 
spends about one-third of its time in the patrol area, 
so that on the average ten days are spent there out of 
each month. During this time the submarine can 
search over an area of 60,000 square miles (assuming 
a sweep rate of 6,000 square miles per day) unless the 
enemy antisubmarine effort is too severe. In U. S. 
submarine attacks against Japanese merchant ves- 
sels, one ship out of eight sighted was sunk. There- 
fore in an area with an average density of shipping of 
one merchant vessel per thousand square miles, an 
operational submarine would sink about eight ships, 
on the average, per operational month. 

Long range aircraft suitable for antishipping work 
average about eighty hours in the air per month. If 
the patrol courses were well laid-out, it would be 
possible to spend a half of this airborne time in the 
shipping area, so that each plane might be expected 
to spend about forty hours per month searching for 
ships. According to Table 2 a reasonable sweep rate 
for merchant vessels might be 500 square miles per 
hour, since merchant vessels are more easily sighted 
than submarines, so that each plane could search 
over about 20,000 square miles each month of opera- 
tion. In an area where there were on the average one 
ship per thousand square miles this plane would 
sight 20 ships per month on the average. Data from 
Coastal Command antishipping planes indicate that, 
with adequate equipment and training, a plane sinks 
about one ship out of every 40 sighted (using bombs 
or rockets). Therefore, in the area under question 
each plane would sink about a half a ship a month. 
Comparing the two, one sees that a single submarine 
is equivalent, as far as sinking enemy shipping goes, 
to a squadron of long range antishipping planes. 


CONFIDENTIAL 


EVALUATION OF EQUIPMENT PERFORMANCE 


51 


A great deal more than this numerical comparison 
must be gone into before it is possible to decide 
whether to use planes or submarines against enemy 
shipping in any given area. In addition to the ques- 
tion of cost of outfitting a submarine as compared to 
the cost of outfitting a squadron of planes, there is 
also the point that the plane can be used for other 
purposes besides sinking ships. The exchange rate 
for the two types of effort must also be taken into 
account. 

3.3.4 Anti-Ship versus Anti-City Bombing 

The quantitative analysis of the relative effective- 
ness of aircraft in bombing the enemy’s factories or 
in sinking his ships takes one still further into ques- 
tions of economics. A possible unit of measure would 
be the monetary value of the destruction caused. 
There is some danger in this, however, for the mone- 
tary value of a building or a ship may be very different 
from its value to a nation at war. The important fac- 
tors in the bombed cities are the destroyed munitions 
or munitions factories; the destruction of housing is 
perhaps not as important, unless it reduces the effi- 
ciency of the munitions workers. Perhaps a better 
unit of measure would be the number of man-months 
required to replace the munitions, rebuild the fac- 
tories, or rebuild the ships sunk. If this could be 
estimated, then it would be possible to compare 
quantitatively an antishipping sortie and a bombing 
sortie over an enemy city, for the relative effective- 
ness in man-months cost to the enemy. 

Work on this general strategic level can only be 
done adequately if the operations research worker has 
access to a great variety of records and intelligence 
reports. In fact it is often impossible to obtain ade- 
quate data on all the important factors from the 
records of one service alone. Unless the worker is 
operating at a high command level, it is usually futile 
to attempt such broad-scale quantitative compari- 
sons. 

3.4 EVALUATION OF EQUIPMENT 

PERFORMANCE 

It has often been said that modern wars are tech- 
nical wars. If this statement has any meaning at all, 
it indicates that new, specialized weapons are de- 
veloped and introduced into operations during the 
war; that we end the war fighting with different 
weapons than we started with. Indeed, in the last 


war there were many cases where the fighting forces 
had not yet learned to use effectively the new weapon 
before that weapon became obsolete. This does not 
mean that an effort should have been made to slow 
down the introduction of new weapons. It means that 
technical thought in learning how and where a new 
weapon should be used, and in teaching the armed 
forces the best use of new equipment, is as important 
as is technical effort in the design and production of 
the new weapons. Here again the quantitative ap- 
proach of operations research can speed up the over- 
all learning time and make it possible to use the new 
weapons effectively before they become obsolete. A 
good bit of this work is not operations research in the 
strict sense of the word; but operations research 
workers should know how to evaluate weapons and 
judge where they are useful, because they are often 
the only technical men on the spot, and must do it. 

3.4.1 First Use of New Equipment 

A great deal of thought must usually be spent on 
the possible tactical use of new equipment before this 
equipment gets into operation. Someone with tech- 
nical knowledge, either in the armed forces, or in an 
operations research group, or in the laboratory, sees 
the tactical need for a new weapon and sees the tech- 
nical means by which this need can be satisfied. If 
this analysis appears to be reasonable a laboratory 
commences development work, production designs 
are gradually worked out and, eventually, produc- 
tion commences. Unfortunately, there are many slips 
between the initial idea and its final confirmation in 
battle. The initial tactical analysis may have been 
faulty, or its embodiment in the new weapon may be 
unworkable, or the tactical situation may have 
changed by the time the new equipment gets into 
operation. It is extremely important therefore, that 
operations research workers, who are in touch with 
the changing tactical situation, take an active part 
in evaluating each stage in the development and pro- 
duction and use of new equipment. It is particularly 
important that the first few operational results with 
the new gear be scrutinized closely to see whether it 
is necessary to improve on the original idea for use 
of the gear. Detailed analysis of the working of the 
equipment must be made so as to devise adequate 
measures of effectiveness for future force require- 
ments, and special action reports must be laid out 
so as to have the operating forces provide data with 
which to compare the operational measures with the 


CONFIDENTIAL 


52 


THE USE OF MEASURES OF EFFECTIVENESS 


theoretical ones. In order to make these comparisons 
and in order to suggest changes in use for improving 
results as rapidly as possible, it is often important to 
send operations research workers near to the front 
to observe as closely as possible the working of the 
new equipment. 

3.4.2 Devising Operational Practice Training 

It usually turns out that the operational forces at 
first have not the necessary training or understanding 
in the use of the new weapon. The operations research 
worker must therefore devise methods whereby the 
fighting forces can learn to use the new weapon while 
they are fighting with it. Practice on the battlefield 
is usually not the most efficient way of learning to 
use a new piece of gear. For rapidity in learning, it is 
necessary that the pupil be scored as rapidly as he 
performs the operation; otherwise he will not re- 
member what he has done wrong if his score turns 
out to be low. Such scoring can seldom be provided 
on the field of battle, where usually the operator can- 
not see the results of his actions. Consequently, a 
practice routine must be set up with means of scoring. 
These scores, of course, are measures of effectiveness 
which can be used to check equipment performance 
as well as rapidity of learning. 

3.4.3 Equipment Evaluation 

At each stage in the development of the new 
weapon from the first idea to its final operational 
embodiment, the operations research worker must 
evaluate its overall usefulness in terms of the follow- 
ing general questions : 

1. Is the new weapon worth while using at all? Is it 
better than some alternative weapon already in use? 
In what way is it better, and is this a different and 
important way? Does the cost of the change pay for 
itself? 

2. When and where should the new weapon be 
used? What are the best tactics for its use, and how 
is this likely to modify the enemy’s tactics? Is the 
weapon easy to counter and if so, what can we do 
about it? How can we find whether the enemy is 
countering or not? 

3. Is the new equipment easy to maintain in oper- 
ation? Are the maintenance crews properly trained, 
and are there understandable maintenance manuals? 
What simple operational tests can be devised to in- 
sure that the equipment is being kept in good repair? 


4. How much training is needed in order that the 
new weapon be more effective than the old one? Can 
the results obtained by the weapon be noticed easily 
in battle, or must operational training, properly 
scored, be carried on continuously to insure effective 
use of the gear? What proportion of operational time 
must be spent in this practice, and how long will it 
take before the fighting forces can use the new 
weapon more effectively than they did the old one? 

Such evaluation is often extremely difficult, partic- 
ularly if the new weapon involves radically new 
principles. Experience gathered close behind the front 
lines is extremely valuable in making such evalua- 
tions. In many cases the evaluation cannot be com- 
plete without supplementing the operational data by 
data collected from operational experiments per- 
formed under controlled conditions. This aspect of 
the problem will be discussed in Chapter 7. 

In the present section Ave will give a number of 
examples of the evaluation of new equipment com- 
ing into operation, showing how some of the ques- 
tions raised above can be answered. 

3.4.4 Antiaircraft Guns for Merchant Vessels 

At the beginning of the war, a great number of 
British merchant vessels were seriously damaged by 
aircraft attacks in the Mediterranean. The obvious 
answer was to equip the vessels with antiaircraft 
guns and crews, and this was done for some ships. 
The program was a somewhat expensive one, how- 
ever, since antiaircraft guns were needed in many 
other places also. Moreover, experience soon showed 
that single guns and crews, with the little training 
which could be spared for merchant vessels, had very 
little chance of shooting down an attacking plane. 
The argument for and against installation had been 
going on for nearly a year with no apparent conclu- 
sions reached. The guns were so ineffective that they 
hardly seemed worth the expense of installation; on 
the other hand, they made the merchant vessel crews 
feel somewhat more safe. In the meantime, opera- 
tional data had been coming in as to the experience of 
ships with and without gun protection, and it was 
finally decided to analyze the data in an attempt to 
settle the question. 

It was soon found that in only about 4 per cent of 
the attacks was the enemy plane shot down. This 
was indeed a poor showing, and seemed to indicate 
that the guns were not worth the price of installa- 
tion. On second thought, however, it became appar- 


CONFIDENTIAL 


EVALUATION OF EQUIPMENT PERFORMANCE 


53 


ent that the percentage of enemy planes shot down 
was not the correct measure of effectiveness of the 
gun. The gun was put on to protect the ship, and the 
proper measure should be whether the ship was less 
damaged if it had a gun and used it, than if it had no 
gun or did not use it. The important question was 
whether the antiaircraft fire affects the accuracy of 
the plane’s attack enough to reduce the chance of the 
ship’s being hit. Figures for this were collected and 
the results are shown in Table 8. It is apparent from 


Table 8. Casualties to merchant vessels from aircraft 
bombing attacks (low-level attacks). 



AA fired 

AA not fired 

Bombs dropped 

632 

304 

Bombs which hit 

50 

39 

Per cent hits 

8 

13 

Ships attacked 

155 

71 

Ships sunk 

16 

18 

Per cent sunk 

10 

25 


this table that for low-level horizontal attacks the 
accuracy of the plane’s attack was considerably re- 
duced when antiaircraft guns were firing, and the 
chance of the ship escaping was considerably better 
when antiaircraft was used. These same results were 
obtained for enemy dive-bombing attacks. It was 
obvious therefore that the installation of antiaircraft 
guns on merchant vessels was something which would 
definitely increase the ship’s chance of survival, even 
though such guns did not shoot down the attacking 
planes very often. 

This numerical analysis finally settled the question. 
For the antiaircraft guns more than paid for them- 
selves if they reduced the chance of the ship being 
sunk by a factor of more than 2. 

3.4.5 Antitorpedo Nets 

Early in the U-boat war in the Atlantic an attempt 
was made to save merchant vessels by equipping 
them with antitorpedo nets which were swung out by 
booms. These nets were capable of stopping some 85 
per cent of the German electric torpedo [G7E] but 
only 20 per cent of the German air-propelled torpedo 
[G7A]. Taking the armament of U-boats as about 
60 per cent G7E and 40 per cent G7A gave an aver- 
age protection against these torpedoes of 59 per cent. 
Since the nets covered only about 75 per cent of the 
ship, the “net” protection was 44 per cent. 

This appears to be a strong argument in favor of 
equipping all merchant vessels with nets. But the 


cost was extremely high and the nets slowed down 
the ships, making an additional cost for fuel and time 
lost. Against some opposition, about 600 ships were 
fitted with nets before enough operational experience 
had been obtained to make a reappraisal possible. 
This reappraisal was quite broad in scope, as it in- 
volved: (1) cost in dollars as against the cost of ships 
saved by the net, (2) cost in time and in cargo space, 
(3) cost in manpower to build and maintain the nets. 
The research on the cost in dollars found that the net 
program did not pay for itself. The operational data 
on the 25 ships which were torpedoed and which were 
fitted with nets are shown in Table 9. If the 10 ships 


Table 9 



Sunk Damaged Undamaged 

12 ships, nets not in use at 
time of attack 

9 

3 

0 

10 ships, nets in use 

4 

3 

3 

3 ships, use of nets unknown 

3 

0 

0 


16 

6 

3 


with nets streamed had not had their nets in use, we 
should expect 7J^ to have been sunk and 2J^ dam- 
aged. The nets had thus saved the equivalent of V /2 
ships and cargoes. But a total of 590 ships had been 
fitted with nets at an initial cost equal to about 
twice that of S }/2 ships and cargo, not to mention 
costs of maintenance, etc. Thus the program had not 
paid for itself, and the report of the findings recom- 
mended that no further ships be equipped with nets. 

The previous two examples were cases where the 
answer was a simple “yes” or “no”; the antiaircraft 
guns were worth the cost of installation, but the anti- 
torpedo nets were not worth the cost. In many other 
cases the answer is not so simple. It may turn out 
that the new weapon has not as general a usefulness 
as was at first thought, but that it is extremely useful 
in certain special circumstances. Here research on the 
measures of effectiveness of the weapon can produce 
a twofold benefit: general efficiency is increased be- 
cause the new weapon is not used in places where it 
is not efficient, and the effectiveness of special oper- 
ations is increased considerably by using the new 
weapon in places where it has a decided advantage. 

3.4.6 Magnetic Airborne Detector 

A clear-cut example of the importance of assigning 
equipment to tasks commensurate with their abilities 
is afforded by the case of the magnetic airborne de- 


CONFIDENTIAL 


54 


THE USE OF MEASURES OF EFFECTIVENESS 


lector [MAD], developed to detect submerged sub- 
marines from aircraft. In addition to being the only 
airborne means for discovering underwater subma- 
rines, MAD offers the advantage of not revealing the 
presence of the aircraft. Although these qualities 
would seem ideal for antisubmarine search, closer 
examination reveals certain features limiting the 
operational effectiveness. 

The advantage of aircraft for visual or radar search 
lies in the high speed plus the broad sweep width 
(roughly twice the detection range) against surfaced 
submarines (up to several miles). For MAD-equipped 
aircraft the sweep rate is so reduced by the low MAD 
detection range (perhaps 100 yards for aircraft at 
250 feet altitude and submarine at 250 feet depth), 
that the expectation of finding a submerged subma- 
rine becomes virtually nonexistent, at least for ran- 
dom search over open ocean areas. It is comparable 
to the efforts of a blind man trying to draw a pencil 
line through a single dot on a large floor. 

Although these characteristics place a definite 
limitation on its usefulness, they do not eliminate 
MAD as a submarine detection device. To the con- 
trary, a full appreciation of the “measures of effec- 
tiveness” of MAD, and its peculiarities, point the 
way to specialized tasks for which it is most effec- 
tive. One such opportunity was exploited during the 
Italian campaign in helping close off the Gibraltar 
Straits to Nazi U-boats enroute to the Mediter- 
ranean. 

U-boats had been making submerged passage of 
the Straits by daylight, utilizing underwater currents 
for propulsion to reduce noise. In the meantime, 
thousands of hours of fruitless MAD search had 
been invested in the North Atlantic, while on the 
other hand, radar search within the Straits was in- 
effective due to the submerged nature of the pas- 
sages. Finally, it was recognized that here was a 
special case where MAD could redeem its record, and 
an MAD patrol, designed in accordance with search 
theory, was instituted across the Straits. Within the 
first two months of operating the MAD barrier across 
the channel, two contacts resulting in U-boat sink- 
ings were obtained, and a third one came soon later. 
The result so discouraged the U-boats that no more 
attempted the passage into the Mediterranean for 
more than six months. Thus even though MAD has 
only between l/50th and l/100th the search rate of 
radar, it was quite capable of providing an effective 
blockade across a restricted area, and one which did 
not provide warning, as would surface craft. 


This is merely one illustration of how the appro- 
priate use of a new device can mean the difference 
between complete failure and success. 

3.4.7 Submarine Torpedo Evaluation 

During World War II the U. S. submarine force 
introduced an electric torpedo, the Mark 18. Previous 
to this, the torpedoes used were the Mark 14 and the 
Mark 23, both steam-propelled. The steam torpedoes 
both ran at 46 knots in their high-power setting. Thus 
they had the advantage of high speed, but they left a 
clearly discernible wake. The electric torpedo ran a 
shorter distance and at a much slower speed, 29 
knots ; however, it had the advantage of being wake- 
less. It was considered that this property of invisible 
travel more than made up for the slower speed, for 
the ship attacked would have no previous warning, 
and the enemy escort vessels would have no torpedo 
track to follow in commencing their counterattack. 
After the new torpedo had been used for some 
months, an evaluation was made to see whether the 
steam torpedoes should not be discontinued entirely. 

In order to answer these questions, a uniform body 
of data was chosen: attacks in a four-month period 
by submarines under a single command. In order not 
to give the Mark 18 torpedo an undue advantage 
(since attacks with it were made at closer range than 
attacks with the Mark 14 and Mark 23), no attacks 
made at ranges over 4,000 yards were considered. 
With these limitations, the analysis of the data re- 
sulted in the following conclusions: 

1. The proportion of successful salvos under equal 
conditions fired against all types of enemy vessels 
(except large combatant units) is greater for the 
Mark 14 and Mark 23 (steam) torpedoes than for the 
Mark 18 (electric). 

2. In attacks on merchant vessels, the proportion 
of successful salvos is greater with the Mark 14 and 
Mark 23 by a factor of 1.14. 

3. In attacks on large combatant units, the propor- 
tion of successful salvos appears to be greater with 
the Mark 18 by a factor of about 1.2. 

4. In attacks on destroyers, escort vessels, and pa- 
trol craft, the proportion of successful salvos is 
greater with the Mark 14 and Mark 23 by a factor of 
1.4, in the case of destroyers, to 2.5, in the case of 
escort vessels and patrol craft. 

5. The occurrence and accuracy of deliberate 
counterattacks by enemy escorts show no correlation 
with the Mark number of the torpedoes fired in 


CONFIDENTIAL 


EVALUATION OF EQUIPMENT PERFORMANCE 


55 


attacks on merchant convoys. This holds true for 
both day and night attacks. 

6. In the case of attacks on warships, the propor- 
tion of enemy counterattacks is, however, somewhat 
smaller with the Mark 18. (It was suggested that this 
might have been due to a better lookout system on 
the large combatant ships.) 

7. It was estimated that, if all U. S. submarines in 
1944 had carried full loads of Mark 18 torpedoes, the 
enemy would have lost about 100 fewer merchant 
ships than if full loads of Mark 14 and Mark 23 had 
been carried. At the same time it was considered 
probable that the exclusive use of the Mark 18 
would not have prevented a single U. S. submarine 
casualty. It was therefore recommended that sub- 
marines use the Mark 14 and Mark 23 torpedoes 
against merchant vessels, and that they use Mark 18 
torpedoes against large combatant units. 

In this case it turned out that the danger against 
which the electric torpedoes were provided (the 
chance that the enemy would see the wake of the 
steam torpedoes) was not as great as had been 
apprehended. This, of course, could not have been 
predicted until a wakeless torpedo had been tried in 
actual operation. It turned out in most cases that the 
reduction in danger to the submarine was negligible, 
but that the loss in accuracy of firing torpedoes, due 
to the slow speed of the electric torpedo, was appre- 
ciable. Luckily the specialized advantage of the 
Mark 18 against large enemy naval vessels was im- 
portant enough to save the whole development pro- 
gram from being a complete waste of effort. 

3 . 4.8 Aircraft Antisubmarine Depth Bombs 

Sometimes an examination of the operational re- 
sults of the first use of new equipment indicates 
clearly that a slight modification of the equipment 
will make it very much more effective. This sort of 
situation has turned up several times in connection 
with the development of the use of aircraft as an 
antisubmarine weapon. The Germans underesti- 
mated the value of aircraft against submarines; in 
the end aircraft played a very important part in the 
defeat of the U-boat in the Atlantic. 

Early in World War II the British Coastal Com- 
mand used ordinary bombs in their attacks against 
submarines. These were obviously not effective, since 
they exploded on the surface of the water, and, if 
they did strike the deck of the submarine, they sel- 
dom penetrated the pressure hull. Depth charges 


were therefore adapted for aircraft dropping. These 
insured an underwater explosion which would be 
considerably more destructive to the submarine hull. 
At this point arose the argument as to what should 
be the depth-setting for the bomb’s explosion under- 
water. It was not possible to change this depth- 
setting in the plane just prior to the attacking run, 
so that an estimate had to be made as to the best 
average setting for all attacks, and this setting had 
to be used all the time. 

A number of squadrons, no doubt feeling that a 
submarine was most likely to be submerged, set their 
depth bombs to explode at 150 feet. The absurdity of 
this setting soon became apparent, however, for sub- 
marines at 150-foot depth could not be seen (and 
therefore not attacked), and submarines near the 
surface which could be seen would only be somewhat 
shaken by an explosion at 150-foot depth. The depth- 
setting was next reduced to 50 feet, as a compromise 
between the “deep setters” and the “shallow setters.” 
After a year of argument, a numerical analysis was 
made which settled the argument. 

The fundamental question was the state of sub- 
mergence of the submarine at the instant the attack- 
ing plane dropped its depth charge. If a great number 
of attacks were made when the submarine was on the 
surface, then the 50-foot depth-setting was too deep, 
for an explosion at such a depth was too far away 
from the pressure hull of a surfaced submarine to 
have a great chance of producing lethal damage. On 
the other hand, if the submarine was in the act of 
diving or had just dived at the instant of attack, then 
perhaps the 50-foot setting would be satisfactory. 

However, attacks after the submarine has dived 
are much less likely to be accurate than attacks on 
surfaced submarines. Therefore, even if the majority 
of attacks were made on submarines which had sub- 
merged a minute or more before the depth charge 
was dropped, it was not sensible to make the setting 
best for these cases, because the chance of success for 
such attacks was very low anyway. The depth- 
setting should be determined by the type of attack 
which had the best chance of success, which was the 
attack on the surfaced submarine (unless it turned 
out that this type of attack was a negligible fraction 
of the total) . 

An examination of the operational results indicated 
that in 40 per cent of the cases the attack was made 
on a surfaced submarine, and in another 10 per cent 
of the cases a part of the submarine was visible when 
the depth charge was dropped. Therefore in half of 


CONFIDENTIAL 


56 


THE USE OF MEASURES OF EFFECTIVENESS 


the cases (the half for which the attack was most 
accurate) the 50-foot depth-setting was too deep. In 
the other half of the cases (when the accuracy was 
considerably less) the 50-foot depth-setting might be 
more satisfactory. A numerical analysis of the 
chances of success of the attack as a function of the 
degree of submergence of the submarine indicated 
that a change in the depth setting from 50 feet to 25 
feet would at least triple the chance of success of the 
average attack. 

In consequence of this analysis, it was made doc- 
trine to set the depth of explosion for aerial depth 
bombs at 25 feet, and to instruct the pilots not to 
drop depth bombs if the submarine had already 
submerged for more than half a minute. Within a 
few months after this change in doctrine went into 
effect, the actual effectiveness of aircraft antisub- 
marine attacks increased by a factor of more than 2. 

3.4.9 The Importance of Maintenance 

Maintenance is a continual problem with modern 
weapons of war. The performance of even the usual 
weapons must be checked from time to time to make 
sure that incomplete care does not seriously reduce 
their effectiveness. As a simple example, we can 
quote (Table 10) from an analysis made of the 


Table 10. Bomb-release failures. 


Type of 
aircraft 

Number of 
incidents 

Bombs failed 
to release 

Failed to 
explode 

PV-1 

40 

4 (10 per cent) 

1 

PBY-5A 

9 

1 (11 per cent) 

0 

PBM 

33 

3 (9 per cent) 

0 

PB4Y 

177 

8 (4.5 per cent) 

0 

Total 

259 

16 (6.2 per cent) 

1 (0.4 per cent) 


mechanical results from bombs dropped by 17 squad- 
rons in the Pacific. Evidently the bomb-fusing mech- 
anism was satisfactory, for only one failed to explode. 
The bomb-release mechanism, however, was not 
satisfactory, for in each 16 attacks there was one 
attack which failed because the bombs hung up. It 
was soon found that this was due to an inadequate 
checking of the bomb shackles. This was soon reme- 
died and the percentage of bomb-release failures 
dropped markedly. 

3.4.10 Height-Finding Radar 

During the last year of the war in the Pacific it be- 
came extremely important to keep enemy suicide 


bombers away from our task forces. Antiaircraft 
equipment on the ships was quite effective, but this 
should, of course, be only used as a last resort. The 
primary defense against suicide bombers is the com- 
bat air patrol [CAP]. When enemy planes are de- 
tected by the search radar a CAP unit is vectored to 
intercept them. This interception is extremely diffi- 
cult unless the unit is given a fairly accurate measure 
of the elevation of the incoming enemy planes. Con- 
sequently, an accurate determination of height by 
the ship’s radar system is an important link in the 
defense of the task force against enemy bombers. 

A detailed analysis of the action reports indicated 
that this height determination gave signs of being the 
weakest link in the defensive pattern. Enemy planes 
were nearly always detected at ranges greater than 
75 miles. CAP units were nearly always vectored to 
intercept, but in entirely too large a fraction of the 
cases no interception resulted. The suspicion that this 
was due to inaccurate height-finding was strength- 
ened by reports that in a number of cases where 
several height-finding radars were in use in the same 
task force they gave discordant results. Experts were 
sent out to several ships having height-finding radars 
installed, and they found a considerable amount of 
inaccuracy in their readings. In a number of these 
cases it was found that the alignment of the antenna 
was out of adjustment, and in a number of other cases 
the operators were not adequately trained. In a num- 
ber of cases the readings were more than 1,000 feet in 
error in elevation, which could easily explain the lack 
of interception of the enemy. 

Here was a difficulty which was a combination of 
poor maintenance and insufficient training. The fun- 
damental error was in not providing a simple and 
scorable test to check maintenance and training at 
frequent intervals. By consulting with the experts 
and with the fleet officers a standard calibration test 
was devised which all ships could conduct in about 
three hours with the use of utility aircraft (or one of 
the ship’s own aircraft) as a target. These tests were 
authorized by the type commander of the theater 
and made it possible for the task force commander to 
test periodically the adequacy of his height-finding 
equipment and operators. 

3.4.11 Radar Bomb Sights 

Occasionally new equipment gets sent into the field 
ahead of manuals or trained personnel, so that the 
theater has little conception of the limitations or pos- 
sibilities of the gear. The operations research worker 


CONFIDENTIAL 


EVALUATION OF EQUIPMENT PERFORMANCE 


57 


in the field can be of considerable help in such cases. 
An example of this occurred when electronic bombing 
equipment was first installed in Navy patrol planes 
in the Pacific. Antishipping strikes with radar bomb- 
ing equipment [APQ-5] in the Pacific areas had not 
been as successful with Navy patrol planes as it had 
been with several Army squadrons (as of June 1, 
1945). These facts were disclosed by a statistical 
survey of the attacks against enemy shipping. 

A study was made to discover the cause of failure 
by examining the equipment performance, the train- 
ing program, and the tactical use of the equipment in 
combat areas. New facts came to light in all three 
categories w r hich promised to solve the difficulty. So 
far as equipment was concerned, it was learned that 
calibration of the gear needed constant attention to a 
degree not appreciated by patrol plane commanders. 
In addition, there was considerable difference in per- 
formance between those planes equipped with the 
auto-pilot (which is connected directly to the radar 
bombing equipment), and other planes where the 
pilot followed the pilot direction indicator [PDI]. 
Thus equipment performance accounted for the 
rather good results of Army (and Navy) Liberator- 
type bombers [ PB4Y-1 ] as against the poor showing 
of Navy PBM-type aircraft. 

In the line of training, the performance of student 
crews was investigated. Each crew was given instruc- 
tion on a ground trainer for four hours and then made 
a flight in a school plane. On this flight the student 
crew watched the instructors drop on the target, and 
then took over and made 3 or 4 drops. The instruc- 
tors, who were an average patrol plane crew singled 
out for the job, were averaging 70 per cent hits, after 
having made 100 bombing runs, whereas the students 
were consistently averaging 35 per cent hits with 
their 3 or 4 drops. Clearly the students were only 
beginning to learn, and the training period was far 
too short. A better return on the investment would 
have resulted if a few crews had been really well 
trained and sent forward to combat areas as special- 
ists. 

Finally, tactics were examined. By July 1, 1945 
the Japanese shipping was moving through open 
waters only at night, and during the day was anchored 
in protected harbors close against land. Our own 
Navy submarines were prowling in enemy waters and 
were surfaced at night, so that doctrine required our 
aircraft to identify positively any vessel attacked as 
a non-sub. There were also areas (submarine sanc- 


tuaries) where aircraft were not allowed to bomb, 
and still other areas where any target could be 
bombed. It was a matter of simple logic to propose 
that the radar bombing equipment be used only at 
night and only in the nonrestricted areas. 

3.4.12 The Effects of Supervised Practice 

A striking example of the far-reaching effects of 
continual, well-supervised practice on the effective- 
ness of operations can be taken from the experience 
with very long range [VLR] strategic bombing of 
Japan. An object of strategic bombing is to destroy 
the enemy’s facilities for producing war goods, with 
special emphasis on those plants engaged in producing 
equipment which is hardest to replace. Many such 
plants are small targets, and to knock them out 
usually requires accurate bombing. Therefore, the 
value of a unit of a strategic air force depends upon 
how many such enemy targets it can destroy in a 
given time. This rate depends on many factors: the 
number of aircraft available, their maintenance, 
the bomb-load per plane, the accuracy with which 
they can drop bombs, etc. 

The training of the bomber crews has an important 
effect on nearly all of these factors. The initial train- 
ing is of course important, but it appears that con- 
tinuous practice-training, which should be carried on 
in the field of operations, is equally important. One 
might expect that an operational crew could practice 
bombing over an enemy target just as well as it 
could practice over a trial target, and it is not ob- 
vious that it would be profitable to reduce the num- 
ber of bombing missions per month in order to allow 
time for continued practice. It is, of course, true that 
scoring of practice bombing is more immediate and 
detailed than the assessment of operational bombing, 
so that the crews can learn more quickly their mis- 
takes. It is also true that more experience can be 
gained in ten hours practice over a trial target than 
in a ten-hour operational mission. 

In order to determine how much value operational 
practice training can be, the data for one YLR com- 
mand was studied for a period of six months. For the 
first half of this period practice bombing was not 
emphasized in this command. During the second 
three-month period somewhat more than 10 per cent 
of the operational time of the crews was spent in 
practice. The curves of Figure 2 show the compara- 
tive results. 


CONFIDENTIAL 


58 


THE USE OF MEASURES OF EFFECTIVENESS 


The topmost of the four curves shows the average 
number of hours a VLR plane of this command spent 
in the air per month. The dotted curve gives the 
total average hours, and the solid curve gives the 
number of hours spent on missions bombing Japan, 
the difference between the curves giving the average 
number of hours per month per plane spent on prac- 
tice. One notices a continual increase in the average 
operational hours per month per plane, indicating 



AVERA 

CAM 

GE PER CE 
E WITHIN 

:nt of BO 
IOOO FT ( 

MBS DROP 
OF POINT C 

PED WHIC 
)F AIM 

H / 














JUNE JULY AUGUST SEPTEMBER OCTOBER NOVEMBER 


Figure 2. Measure of effectiveness of VLR planes for 
aimed bombing to show effects of operational practice 
training. (Data for one command.) 


that by November each plane was working approxi- 
mately 70 per cent more time than the same plane 
had in June. This improvement is only to a small part 
due to the training of air crews, the principal contri- 
butions having been due to increased experience of 
the maintenance personnel, to modifications in the 
aircraft, and to cooler weather. One sees that during 
and after September a considerably larger amount of 
time was spent in practice. 

The second curve shows the improvement in load- 
carrying capacity of the plane. A large portion of the 
phenomenal rise during September and October was 


due to the standardization of “stripping and weigh- 
ing” the aircraft. Another contributing factor was 
the increased experience on the part of pilots, flight 
engineers, and navigators, so that the cruise-control 
data were more closely followed and less fuel reserve 
was required. A detailed analysis indicates that the 
increase in average bomb-load was in part due to the 
additional training, that a more experienced crew 
can, in general, carry a larger load. 

The third curve shows a measure of the bombing 
accuracy of the plane, obtained from assessment of 
photographs of damage. What is plotted is the aver- 
age per cent of the bombs dropped which come 
within 1,000 feet of the target. It is to be noted that 
the upward trend in the bombing accuracy begins 
shortly after the increase in flying training shown in 
the top curve. Other factors affecting the accuracy 
were changes in formation, permitting the bombings 
to be controlled by a small number of lead crews, 
which are given additional training. Improved 
weather, with resulting better physical conditions 
for the crews, probably also had its effect; though, if 
this was an important factor, the rise should have 
occurred in September and early October rather than 
later. One can conclude that the rise in accuracy is 
due in no small part to increase in training. 

The product of these three factors (hours per 
month per plane, bomb-load carried, and accuracy) 
could be used as an overall measure of effectiveness 
of an individual VLR plane and crew for the strategic 
bombing of small targets. This product is plotted in 
the lower curve of Figure 2. The number of hours per 
month is, of course, the number of hours of actual 
bombing (i.e., the total number of hours per month 
minus the number of hours used in bombing prac- 
tice). This is the solid curve, called “true effective- 
ness.” We notice very little change in this measure 
throughout the first three months. From the time 
the additional training was instituted to the end of 
the three months’ period, there is, however, a phe- 
nomenal rise. Each plane, by the end of November, 
is approximately ten times as effective as the same 
plane and crew was the first of September. 

As has been indicated, some of the rise is due to 
weather and other causes, but the most important 
cause seems to have been the additional practice 
training. As a conservative estimate of the cost of 
this training, we can use the total hours per month 
per plane instead of the actual bombing hours. This 
gives rise to the dotted curve of the lower graph, 


CONFIDENTIAL 


EVALUATION OF EQUIPMENT PERFORMANCE 


59 


which is the measure of effectiveness which a plane 
would have if it had spent all its operational hours 
in bombing targets, and if the other improvements 
(of bomb load carried and of accuracy) had occurred 
without the training. We see that the loss of effec- 
tiveness due to the additional time taken out in 
training is a very small amount compared to the 
extraordinary rise in bomb-load and accuracy, which 
the training in part produced. 

The really large change in effectiveness, as shown 
in the fourth graph, is of considerable interest. It 
means, in fact, that a VLR plane was ten times as 
effective in destroying enemy installations in Novem- 
ber as in August. In other words, a squadron of these 
planes was more effective in November than three 
groups were in August, though the planes were essen- 
tially the same. These figures show the futility of 
numbers compared to quality of performance. The 
results achieved by increased efficiency (by further 
training) obviously far exceeded those which could 
have been reached simply by increasing the number 
of aircraft or the number of missions flown, or by 
adding more new gadgets. This fact is often over- 
looked in an attempt to win by numbers, with every 
thing being sacrificed simply for more numbers, more 
aircraft, more sorties, more bombs dropped. The 
curve shows how little mere quantity can count com- 
pared to improved quality. Furthermore, the biggest 
improvements, those of plane-loading and accuracy, 
are largely due to training and continual practice. 

In other aspects of warfare the effect of practice 
and training may not be so decisive, but in all cases 
heretofore analyzed they have turned out to be ex- 
ceedingly important. It would be worth while analyz- 
ing other cases in detail, so that in the future one 
might be able to estimate whether the addition of 
new equipment or the further training of personnel 
in the use of the old equipment would be more effec- 
tive in a given case. 

3.4.13 Evaluating the Enemy’s 

Countermeasures 

In a few cases equipment is misused or discarded 
because of an exaggerated fear of the vulnerability of 
the gear to enemy countermeasures. Such a possi- 
bility must always be prepared for, but it was the 
experience in World War II that our forces usually 
credited the enemy with an effective countermeasure 
long before it actually occurred. This subject has 


already been mentioned, and will be discussed in 
some detail in Chapter 5. An example will be given 
here how a calculation of average effectiveness can 
serve to answer the fears of enemy countermeasures, 
and to prolong the use of an effective piece of equip- 
ment. The example concerns the use of aircraft- 
warning radar by U. S. submarines. The advantage 
of such equipment is that it will detect the aircraft at 
greater range than will visual lookouts, and thus will 
give the submarines a better chance of diving before 
the attack is made. The first early-warning set in- 
stalled on U. S. submarines had an average range of 
detection for Japanese planes only 1.4 times the 
average range of visual detection for these planes; 
nevertheless, this added modicum of warning proved 
of considerable value in a number of cases. In fact, 
our submarines were caught on the surface by the 
Japanese planes in less than 5 per cent of the attacks, 
as compared to the 50 per cent chance of surface 
attack enjoyed by our planes against the U-boat, as 
mentioned earlier. 

In the course of radar development, however, the 
Allies designed and built a radar detector which 
could pick up the signals from a submarine’s early- 
warning radar. It was feared that, if Japanese planes 
had such a radar receiver on them, the range of de- 
tection of submarines using radar, by such planes, 
would be greatly increased, and that the submarine’s 
radar, instead of being an asset, would become a 
liability. Many submarine skippers became con- 
vinced that this had occurred, for they seemed to 
find that more Japanese planes came into view when 
their early-warning radar was turned on than when 
it was off. The operational data were analyzed to see 
if this were so. The results of this analysis are given 
in Table 1 1 . 


Table 11. Aircraft contacts by U. S. submarines per 
100 days stay of submarine in area. 



Area A 

Area B 

Radar early-warning in use 

86 

67 

Radar early-warning not in use 

61 

51 

Ratio 

1.4 

1.3 


These results indicate that submarines with their 
radar in use saw more planes per hundred days than 
those submarines with radar not in use. At first 
sight, therefore, it would seem to indicate that the 
radar-using submarines were attracting planes. On 


CONFIDENTIAL 


60 


THE USE OF MEASURES OF EFFECTIVENESS 


second thought, however, one sees that this is not so. 
The radar-using submarine should see more planes 
because its radius of detection is greater than for 
submarines where visual sighting is relied upon. In 
fact, the ratio of the numbers of aircraft contacts 
should be equal to the ratio of the radii of detection, 
which, as we have seen above, is just the factor 1.4. 
Consequently the operational data indicate that the 
radar-using submarines have seen no more planes 
than they should, compared to the vision-using sub- 


marines. In other words, no more Japanese aircraft 
congregated over radar-using submarines than con- 
gregated over non-radar-using submarines, and, if 
the Japanese had a radar receiver, it was not doing 
them any good. This analysis helped to kill opposi- 
tion to the use of radar in U. S. submarines. Since 
the end of the war, it has been learned that the Ja- 
panese had considered installing radar receivers on 
their planes, but the proposal had been vetoed by 
the higher command. 


CONFIDENTIAL 


Chapter 4 

STRATEGICAL KINEMATICS 


I n chapter 3 we showed that in a great number of 
cases the approximate constants of warfare are 
useful without further mathematical analysis, once 
they have been obtained by theoretical or statistical 
methods. In Section 3.1 we gave an example from 
the antisubmarine war in the Bay of Biscay, which 
showed that sometimes the comparison of the value 
of the constants obtained from operational data with 
the theoretically possible value will indicate, without 
further analysis, the modification in tactics necessary 
to obtain improvement. Similarly, we shall see in 
Section 5.1 that a simple statistical analysis of the 
constants entering into the effectiveness of suicide 
bombers against maneuvering surface craft is all that 
is necessary to indicate the best tactics to be used in 
avoiding “Kamikazes.” Once in a while, a simple 
comparison of two measures of effectiveness will 
suffice to answer a strategical question, such as the 
case discussed in Section 3.3 concerning the relative 
advantage of aircraft and submarines in sinking 
enemy shipping. Measures of effectiveness, statis- 
tically or analytically determined, can be of con- 
siderable aid to the strategic planner in working out 
force requirements for various tasks. This chapter 
will give examples of the strategic usefulness of some 
of these constants, and will indicate some of the 
mathematical theory which is basic to strategic ap- 
plications of the constants. 

4.1 FORCE REQUIREMENTS 

Two examples from the antisubmarine war in the 
Atlantic will suffice to indicate how measures of 
effectiveness can aid in determining force require- 


Miles 

from 

Average number of units present 
Indepen- Naval 

base 

Convoys 

dents 

vessels 

Tugs 

0-100 

4.3 

17.4 

2.0 

1.4 

100-200 

2.1 

2.3 

0.9 

0.3 

200-300 

0.6 

1.7 

0.7 

0.1 

300-400 

0.3 

0.3 

0.2 

0 

400-500 

0.1 

0.1 

0.1 

0 


ments. The first shows the method of calculating the 
number of antisubmarine patrol planes needed in a 
sea frontier, in order adequately to escort the ship- 


ping in the frontier. At one period in World War II, a 
certain frontier had a density of shipping off its coast 
as shown in the foregoing table. 

The ocean area in the sea frontier was divided up 
into zones at different distances from air bases. 
Shipping charts were then counted, so as to obtain 
the average number of naval units, convoys, etc., 
which were to be found each day in each of these 
zones. For instance, on the average, there turns out 
to be about one and two-thirds independent vessels 
in the zone between 200 and 300 miles away from an 
air base; there turns out to be one-tenth of a convoy, 
on the average, in the zone between 400 and 500 
miles from an air base (or, rather, there is a convoy 
in this region one-tenth of the time). 

The region patrolled by the sea frontier planes is 
divided in this way because it takes more effort to 
patrol at large distances from a base than it does at 
short distances. Or, to put it another way, it takes 
more planes to give a convoy adequate coverage 
when the convoy is far from the air base than when 
it is close. The next part of the problem, therefore, 
is to determine how many planes of a given type are 
needed to cover a single unit continuously in a given 
zone. Each plane can fly so many hours a month; the 
rest of the time it must be at the base in order to rest 
its crew, and to undergo overhauling. Suppose the 
average number of hours a month a certain plane 
can be operational is N . Each plane has a maximum 
number of hours T during which it can stay aloft; 
this can be called the length of mission. Not all this 
length of mission is available for escorting vessels, 
however: the plane must fly from the base out to the 
position of the unit before it can be of use, and it 
must fly back again to be of use next time. This 
transit time is equal to twice the distance to the 
center of the zone in question, divided by the speed 
of the plane : 

L = Transit time = ^ (100 + 2 D) , 

where V is equal to the speed of plane in knots, and 
D is the distance from the base to the inner edge of 
the zone in question. 


CONFIDENTIAL 


61 


62 


STRATEGICAL KINEMATICS 


4.1.1 Requirements for Air Escort 

The length of time which a plane can devote to 
convoying on each mission is therefore T — L. There- 
fore, the fraction (T — L)/T is the portion of each 
mission which is actually spent in convoying, so that 
N times this fraction is the total number of hours a 
month a single plane can spend in actual escort of a 
unit in the zone in question. Therefore, the number 
of planes of a given type required to be kept on hand 
at a base in order to provide continuous escort of a 


From this set of tables of force requirements, one 
can calculate the total number of planes required, as 
soon as one knows the particular plane which is to 
cover a given zone, and as soon as one decides what 
percentage of coverage each unit is to be given. For 
instance, if one wishes to give all convoys complete 
coverage, naval vessels 50 per cent coverage, and 
independents and tugs 10 per cent coverage; and if 
one is to use Liberators for the outer two zones, 
blimps for the next two zones and for half the cover- 
age in the inner zone, and Kingfishers for the other 



Blimp 

ZP 

Kingfisher 

OS2U 

Liberator 

PB4Y 

N, hours per plane per month 

360 

80 

100 

V, speed in knots 


50 

90 

130 

T, length of mission, hours 


18 

4 

15 

Distance from bases 





0-100 

2.25 

12.46 

7.58 

Number of planes re- 

100-200 

3.00 


8.49 

quired for full escort 

200-300 

4.50 


9.66 

of one unit 

300-400 

9.00 


11.22 


400-500 • 



13.38 


single unit in a given zone is determined by the 
equation (assuming 30 days per month) : 

Number of planes required at base = — — 

N(T - L) 

Typical performance figures for two different types of 
patrol planes and for navy blimps, together with the 
number required of each type to give continuous 
coverage at different distances from the base, are 
given in the above table. 

We see from the table that the Kingfishers [ OS2U] 
can only be used for cover in the first zone, and the 
Liberators are the only planes mentioned in this 
table which can cover the outer zone. We also see 
that it takes nearly twice as many Liberators to cover 
a unit in the outermost zone as it does for Liberators 
to cover a unit in the innermost zone. 

These two tables can be combined to give the total 
force requirements for complete coverage of the 
different units present in the sea frontier in question, 
as shown in the table on page 63. One multiplies the 
number of planes required per unit in a given zone by 
the number of units present in that zone. This gives 
the number of planes or blimps of each kind which 
would be required to be on hand at bases in order to 
cover all the units present in the frontier all the time. 


half of the coverage of the inner zone; then one sees 
that one needs about 6 Liberators, about 22 blimps, 
and about 45 Kingfishers on base in order to satisfy 
these requirements for close coverage against sub- 
marines. More than this number would need to be 
on hand in order to provide against simultaneous 
breakdown, but this is another problem, already 
touched on in Section 2.2. Other similar requirements 
must be made up for other antisubmarine duties, such 
as for general patrols and to take part in submarine 
hunts after a contact has been made. 

It should be pointed out that a certain per cent of 
nonflying weather simply lowers the number of hours 
per plane per month, but does not affect the aircraft 
assignments made in the tables above. Hours lost due 
to bad weather are hours lost, and the requisite num- 
ber of planes must be present to take advantage of 
the good weather. 

4.1.2 Expenditure of Depth Charges 

Another example of force requirement calculations 
is given in the determination of the depth charges 
and ahead-thrown charges used per month in the 
Atlantic in antisubmarine warfare in 1944. During 
this time there were approximately 30 enemy sub- 
marines on patrol in the Atlantic, and there were 


CONFIDENTIAL 


LANCHESTER’S EQUATIONS 


63 


Aircraft Requirements for A/S Escort for Sea Frontier 


Day and night cover for convoys 


Zone 

Avg No. units 
to be covered 

Blimps 

ZP 

Kingfishers 

OS2U 

Liberators 

PB4Y 

0-100 

4.3 

9.7 

53.6 

32.6 

100-200 

2.1 

6.3 

Cannot 

17.8 

200-300 

0.6 

2.7 

cover 

5.8 

300-400 

0.3 

2.7 

these 

3.3 

400-500 

0.1 


zones 

1.3 


Day and night cover for naval vessels 


Zone 

Avg No. units 
to be covered 

Blimps 

ZP 

Kingfishers 

OS2U 

Liberators 

PB4Y 

0-100 

2.0 

4.5 

25 

15 

100-200 

0.9 

2.7 

Cannot 

7.6 

200-300 

0.7 

3.1 

cover 

6.8 

300-400 

0.2 

1.8 

these 

2.2 

400-500 

0.1 


zones 

1.3 


Day and night cover for independents and tugs 


Zone 

Avg No. units 
to be covered 

Blimps 

ZP 

Kingfishers 

OS2U 

Liberators 

PB4Y 

0-100 

18.8 

42 

234 

142 

100-200 

2.6 

7.8 

Cannot 

22 

200-300 

1.8 

8.1 

cover 

17 

300-400 

0.3 

2.7 

these 

3.3 

400-500 

0.1 


zones 

1.3 


about 500 antisubmarine ships at sea in the Atlantic, 
which used 614 depth charges and 700 ahead-thrown 
charges per month to sink, on the average, 1.25 sub- 
marines a month. It seems to have turned out that 
the number of depth charges and ahead-thrown 
charges used per month is proportional to the num- 
ber of enemy submarines present at any time. For 
the year 1944, therefore, about 20 depth charges 
and 23 ahead-thrown charges were used per month 
per enemy submarine present. This figure was used 
to predict the number of weapons needed in subse- 
quent months once it was possible to estimate the 
number of enemy submarines which were likely to 
be on patrol. This result was used in determining the 
production orders for the successive year. 

It is possible, on the other hand, that the number 
of depth charges used was proportional to the num- 
ber of submarines sunk; this figure for 1944 was about 
490 depth charges and 560 ahead-thrown charges per 
German U-boat sunk. From Intelligence reports, one 


can estimate the number of submarines that will be 
in the Atlantic at some future time, and find the 
number of submarines expected to be sunk. This 
again gave an alternate estimate of requirements, 
which turned out to agree with the other estimate 
approximately. 

4.2 LANCHESTER’S EQUATIONS 

The previous section gave a few simple examples of 
the use of measures of effectiveness to determine 
force requirements. As usual, the constants of war- 
fare are not very constant, and only approximate 
forecasts can be obtained. In most cases such con- 
stants are not known sufficiently accurately to war- 
rant their being used in mathematical equations of 
any complexity. 

Occasionally, however, it is useful to insert these 
constants into differential equations, to see what 
would happen in the long run if conditions were to 


CONFIDENTIAL 


64 


STRATEGICAL KINEMATICS 


remain the same, as far as the constants go. These 
differential equations, in order to be soluble, will have 
to represent extremely simplified forms of warfare; 
and therefore their range of applicability will be 
small. We shall point out later in this chapter other 
serious limitations of such equations. Nevertheless, 
it sometimes happens that considerable insight can be 
obtained into the interrelationship between measures 
of effectiveness by studying differential equations 
involving them. Most of these equations compare the 
losses of the opposing forces, and are obviously re- 
lated to the corresponding equations for chemical 
reactions or for the biological increase or decrease of 
opposing species. A great number of different equa- 
tions of this general sort can be set up, each corre- 
sponding to a different tactical or strategical situa- 
tion, and only a few of them having more than mar- 
ginal interest. A few of the more interesting examples 
will be given in the present chapter, more as indica- 
tions for directions of further investigations rather 
than as descriptions of methods of proven utility. 

4.2.1 Description of Combat 

Some of the simplest and most interesting differ- 
ential equations relating opposing forces were studied 
by Lanchester during World War I . 9 The following 
material is taken from his work. 

One of the great questions at the root of all strategy is that 
of concentration; the concentration of the whole resources of a 
belligerent on a single purpose or object, and concurrently the 
concentration of the main strength of his forces, whether naval 
or military, at one point in the field of operations. But the 
principle of concentration is not in itself a strategic principle; 
it applies with equal effect to purely tactical operations; it is on 
its material side based on facts of purely scientific character. 

There is an important difference between the methods of 
defence of primitive times and those of the present day which 
may be used to illustrate the point at issue. In olden times, 
when weapon directly answered weapon, the act of defence was 
positive and direct, the blow of sword or battleaxe was parried 
by sword and shield; under modern conditions gun answers 
gun, the defence from rifle-fire is rifle-fire, and the defence 
from artillery, artillery. But the defence of modern arms is 
indirect: tersely, the enemy is prevented from killing you by 
your killing him first, and the fighting is essentially collective. 
As a consequence of this difference, the importance of concen- 
tration in history has been by no means a constant quantity. 
Under the old conditions it was not possible by any strategic 
plan or tactical maneuver to bring other than approximately 
equal numbers of men into the actual fighting line; one man 
would ordinarily find himself opposed to one man. Even were 
a general to concentrate twice the number of men on any given 
portion of the field to that of the enemy, the number of men 
actually wielding their weapons at any given instant (so long 


as the fighting line was unbroken) was, roughly speaking, the 
same on both sides. Under present-day conditions all this is 
changed. With modern long-range weapons — fire-arms, in 
brief — the concentration of superior numbers gives an imme- 
diate superiority in the active combatant ranks, and the 
numerically inferior force finds itself under a far heavier fire, 
man for man, than it is able to return. The importance of this 
difference is greater than might casually be supposed, and since 
it contains the kernel of the whole question, it will be exam- 
ined in detail. 

In thus contrasting the ancient conditions with the modern, 
it is not intended to suggest that the advantages of concen- 
tration did not, to some extent, exist under the old order of 
things. For example, when an army broke and fled, undoubt- 
edly any numerical superiority of the victor could be used with 
telling effect, and, before this, pressure, as distinct from blows, 
would exercise great influence. Also the bow and arrow and the 
crossbow were weapons that possessed in a lesser degree the 
properties of fire-arms, inasmuch as they enabled numbers 
(within limits) to concentrate their attack on the few. As here 
discussed, the conditions are contrasted in their most accen- 
tuated form as extremes for the purpose of illustration. 

Taking, first, the ancient conditions where man is opposed to 
man, then, assuming the combatants to be of equal fighting 
value, and other conditions equal, clearly, on an average, as 
many of the “duels” that go to make up the whole fight will go 
one way as the other, and there will be about equal numbers 
killed of the forces engaged; so that if 1,000 men meet 1,000 
men, it is of little or no importance whether a “Blue” force of 
1,000 men meet a “Red” force of 1,000 men in a single pitched 
battle, or whether the whole “Blue” force concentrates on 500 
of the “Red” force, and, having annihilated them, turns its 
attention to the other half; there will, presuming the “Reds” 
stand their ground to the last, be half the “Blue” force wiped 
out in the annihilation of the Red force a in the first battle, and 
the second battle will start on terms of equality — i.e., 500 
Blue against 500 Red. 

4 2.2 The Linear Law 

To set the discussion into a mathematical equation, 
we will let m be the number of combatants in the Red 
force at any instant and n be the corresponding num- 
ber in the Blue force. The time variable in the equa- 
tions requires a little explanation, since it is very 
seldom that warfare goes on continuously. In the 
simplified picture of earlier warfare, each engagement 
(or charge, or battle) was made up of a large number 
of individual combats (or duels). We can label each 
engagement in sequence and use the indicial number 
as the “time” variable t, by the usual extension from 
discrete to continuous variable. Or else we can label 
the individual combats in sequence and use this index 
for our time variable T. 


a This is not strictly true, since towards the close of the 
fight the last few men will be attacked by more than their own 
number. The main principle is, however, untouched. 


CONFIDENTIAL 


LANCHESTER’S EQUATIONS 


65 


We will only consider those combats which result 
in the elimination of one or the other combatant. To 
make the discussion general, we can allow one side 
or the other a certain superiority in weapons or in 
training which can be represented in terms of an 
exchange rate. As explained before, this is the ratio 
E between the average number of Blue combatants 
lost to the average number of Red combatants lost. 
The number of the Red forces lost per combat is 
equal, on the average, to the ratio of the losses in- 
flicted by the Blue forces on the Red and the total 
number of combats (which is equal to the total num- 
ber of losses); and similarly for the Blue losses. 
Therefore the differential equations for the changes 
in m and n per combat are 

dm _ 1 dn _ E 

dT~ ~ 1 + E’ dT~ ~ l + E’ 


m = ra 0 — 


T 

l + E’ 


n = n 0 — 


TE 

l + E’ 


where 


— = E , no — n = E(mo — m) . (1) 

dm 


Since the solutions are linear in T and since the rela- 
tionship between m and n is linear, this set of equa- 
tions is sometimes called Lanchester’s linear law. 

To express the equations in terms of t, we can 
assume that in the £th engagement there are F(m, 
n, t) combats. The equations in terms of the “en- 
gagement variable” t are therefore 


dm 

dt 


1 

1 + E 


F(m, n, t ) , 


dn 

dt 


E 

1 + E 


F(m, n, t) . 


( 2 ) 


Dividing one of these equations by the other, we 
obtain again ( dn/dm ) = E, as before. 

The solutions of these two equations represent 
average or expected values in the sense of probability 
theory. The actual results of any series of engage- 
ments will deviate from this average according to the 
probability analysis given in the next section. 

We see that the solutions to these equations cor- 
respond to the situation discussed above by Lan- 
chester. The two opposing forces are equally balanced 


if the ratio of their initial numbers is equal to the ex- 
change rate E, as has been mentioned above. There 
is consequently no advantage in concentration of 
forces. 


4.2.3 The Square Law 

When we turn to the modern case with extended 
fire power, we find that we cannot break up the indi- 
vidual engagements into unit combats, for each par- 
ticipant in an engagement can fire at every opponent 
(at least in the ideal case). The time variable must 
therefore be the indicial number t of the engagement. 
We will assume that in th £th engagement, a single 
Red combatant can put out of action (E/1 + E)G(t) 
Blue combatants, on the average, and an individual 
Blue combatant can put out of action (1/1 + E)G(t) 
Red combatants, on the average, where G measures 
the “intensity of the combat” at the time t. The cor- 
responding differential equations for this case are 
therefore 


dm _ n . dn 
dt l + E dt 


mE 
1 + E 


Off); 


dm _ n 
dn mE ’ 


n 0 2 — n 2 = E(m 0 2 — m 2 ) ; 


( 3 ) 


where E is again the exchange rate. Since the solu- 
tion of this equation comes out as a relationship be- 
tween the squares of the numbers of the combatants, 
this equation is sometimes referred to as Lanchester’s 
square law. 

The advantages of concentration are apparent in 
the solution of equations (3), for it is apparent that 
the effective strength of one side is proportional to 
the first power of its efficiency and proportional to 
the square of the number of combatants entering the 
engagement. Two opposing forces are then equally 
matched when the exchange rate is equal to the 
square of the ratio of the number of combatants. Con- 
sequently, it is more profitable to increase the number 
of participants in an engagement than it is to increase 
(by the same amount) the exchange rate (by increas- 
ing the effectiveness of the individual weapons) . This 
is not an argument against increased weapon effi- 
ciency; it is simply a statement that a tactical or 
strategical use of concentration may counterbalance 
any moderate advantage in weapon efficiency. 

To bring this fact out more clearly, we will return 
to the engagement mentioned earlier between 1,000 


CONFIDENTIAL 


66 


STRATEGICAL KINEMATICS 


men on the Blue side and 1,000 men on the Red side, 
each with weapons of equal firepower (E = 1). If each 
side throws in all its manpower into each engagement, 
the series of battles will end in a draw. If, however, 
the Red general maneuvers so as to bring his 1,000 
men into engagement with half of the Blue force, it 
will be seen that the Blue force is wiped out of exist- 
tence with a loss of only about 134 men of the Red 
force, leaving 866 to meet the remaining 500 of the 
Blue force with an easy and decisive victory. The 
second engagement between 866 Red participants 
and 500 Blue will result in the annihilation of the 
second Blue contingent with the loss of about 159 
Reds, leaving 707 survivors. 

4.2.4 Fighting Strength 

These equations and their solutions have a great 
range of approximate application and suggest a num- 
ber of useful investigations. Indeed, an important 
problem in operations research for any type of war- 
fare is the investigation, both theoretical and statis- 
tical, as to how nearly Lanchester’s laws apply. If 
it turns out that Lanchester’s square law applies, the 
possibilities of a concentration of forces should at 
once be studied. An obvious application is in aerial 
warfare. It has already been mentioned that an im- 
portant factor in the large ratio of effectiveness 
between U.S. fighting planes and Japanese fighting 
planes lies in the fact that the U.S. planes fight in 
groups of two or three, whereas the Japanese planes 
usually fight singly. 

Another quotation from Lanchester 9 is of interest 
here: 

It is easy to show that this solution may be interpreted more 
generally; the “fighting strength” of a force may be broadly 
defined as proportional to the square of its numerical strength 
multiplied by the fighting value of its individual units. 

As an example of the above, let us assume an army of 
50,000 giving battle in turn to two armies of 40,000 and 30,000 
respectively, equally well armed; then the strengths are equal, 
since (50,000) 2 = (40,000) 2 + (30,000) 2 . If, on the other hand, 
the two smaller armies are given time to effect a junction, then 
the army of 50,000 will be overwhelmed, for the fighting 
strength of the opposing force, 70,000 is no longer equal, but 
is in fact nearly twice as great — namely, in the relation of 49 
to 25. Superior morale or better tactics or a hundred and one 
other extraneous causes may intervene in practice to modify 
the issue but this does not invalidate the mathematical state- 
ment. 

Let us now take an example in which the difference in the 
fighting value of the unit is a factor. We will assume that, as a 
matter of experiment, one man employing a machine-gun can 
punish a target to the same extent in a given time as sixteen 


riflemen. What is the number of men armed with the machine- 
gun necessary to replace a battalion a thousand strong in the 
field? Taking the fighting value of a rifleman as unity, let n = 
the number required. The fighting strength of the battalion is 
(1,000) 2 or 

1,000,000 = 1,000 = 250 

16 4 

or one quarter the number of the opposing force. 

This example is instructive; it exhibits at once the utility 
and weakness of the method. The basic assumption is that the 
fire of each force is definitely “concentrated” on the opposing 
force. Thus the enemy will concentrate on the one machine- 
gun operator the fire that would otherwise be distributed over 
four riflemen, and so on an average he will only last for one 
quarter the time, and at sixteen times the efficiency during his 
short life he will only be able to do the work of four riflemen in 
lieu of sixteen, as one might easily have supposed. This is in 
agreement with the equation. The conditions may be regarded 
as corresponding to those prevalent in the Boer War, when 
individual-aimed firing or sniping was the order of the day. 

When, on the other hand, the circumstances are such as to 
preclude the possibility of such concentration, as when search- 
ing an area or ridge at long range, or volley firing at a position , 
or “into the brown,” the basic conditions are violated, and the 
value of the individual machine-gun operator becomes more 
nearly that of the sixteen riflemen that the power of his weapon 
represents. The same applies when he is opposed by shrapnel 
fire or any other weapon which is directed at a position rather 
than the individual. It is well thus to call attention to the 
variations in the conditions and the nature of the resulting 
departure from the conclusions of theory; such variations are 
far less common in naval than in military warfare; the indi- 
vidual unit — the ship — is always the gunner’s mark. 

Apart from its connection with the main subject, the present 
line of treatment has a certain fascination, and leads to results 
which, though probably correct, are in some degree unex- 
pected. If w r e modify the initial hypothesis to harmonize with 
the conditions of long-range fire, and assume the fire concen- 
trated on a certain area known to be held by the enemy, and 
take this area to be independent of the numerical value of the 
forces, then we have the conditions of equations (1), and the 
rate of loss is independent of the numbers engaged, being di- 
rectly as the efficiency of the weapons. Under these conditions 
the fighting strength of the forces is directly proportional to 
their numerical strength; there is no direct value in concentra- 
tion, ‘qua’ concentration, and the advantage of rapid fire is 
relatively great. Thus in effect the conditions approximate 
more closely to those of ancient warfare. 

4.2.5 Mathematical Solution 

The detailed solution of Lanchester’s square law 
has been studied by Koopman. 10 In order to simplify 
equation (3) one can make a transformation of the 
time variable into 

'-I Tif am - <4) 



CONFIDENTIAL 


PROBABILITY ANALYSIS OF LANCHESTER’S EQUATIONS 


67 


Since the time scale is rather arbitrary in any case, 
the new variable will be just as satisfactory as the 
other one. In terms of the new variable the equations 
and their solutions reduce to the following: 


dm 

dr 


Ve’ 


dn 

dr 


— my/E ; 


the actual case, on the other hand, there is a certain 
small but nonzero chance that all of one side will be 
eliminated without the loss of any combatants on the 
other side, and so on, for all possible proportions of 
losses. The probabilities of the various outcomes can 
be computed if we assume that the results of each 
engagement are subject to the laws of probability. 


m = m 0 cosh r — (n Q / y/E) sinh r ; 


(5) 


4.3.1 


The Linear Law 


n = n 0 cosh r — (m 0 y/ E) sinh r . 

An interesting property of these hyperbolic solutions 
is the acceleration of the action toward the end. The 
last half of the weaker force is annihilated in a shorter 
time than is the first half. This is, of course, due to 
the fact that the remaining members of the stronger 
force are able to concentrate their fire entirely upon 
the remnant of the weaker force, thus accelerating 
the destruction. This acceleration is often apparent 
in actual warfare. One might mention the German 
collapse in Tunisia, and the Allied-versus-German 
air struggle in 1943-44. 

The next two sections present a detailed discussion 
of Lanchester’s equations and related analyses, which 
are in the nature of footnotes, indicating possible di- 
rections for further investigation, rather than results 
which have been useful up to the present. They may 
be by-passed without loss of background for later 
material. 


4.3 PROBABILITY ANALYSIS OF 
LANCHESTER’S EQUATIONS 

Lanchester’s equations deal with the average, or 
expected, values of the number of combatants on 
each side. Actually probability enters, and the results 
of Lanchester’s equations are simply the most prob- 
able results. For the first stages of the battle there is, 
of course, a certain finite probability of other num- 
bers of combatants surviving, and in the later stages 
of the battle the solutions to Lanchester’s equations 
may deviate widely from the possible results. This is 
due to the fact that after a certain length of time 
there is a certain probability that all of one side will 
have been eliminated, and the battle will actually 
have been terminated before the average solution of 
Lanchester’s^ equations predicts that it would end. 

In other words, Lanchester’s equations predict 
that when one side has been completely eliminated, a 
definite number of the other side always remain. In 


For instance, for the linear equations, we can say 
that at each combat, on the average, (E/E + 1) Red 
units are lost, and, on the average, (1/E + 1) Blue 
units are lost. Then, after T combats (if T is smaller 
than n 0 or m 0 ), the multinomial distribution shows 
that the probability that there will be a Red units 
lost and (3 (= T — a) Blue units lost is 

p( “’ / 3 )= ^!(TT^ ;7T = “ + ^ Wo ’” o; (6) 

so that a wide range of outcomes are possible, some 
of them differing widely from the solutions of equa- 
tions (1). However, for a given time T (less than 
ra 0 or n 0 ) the average number of Red and Blue units 
lost is just that given in equations (1). Therefore, for 
the first part of the engagement, the solution to Lan- 
chester’s equation is valid, on the average. 

When the index T gets large enough, however, 
there is a chance that all of one force is annihilated. 
For instance, when T = n 0 , there is a certain proba- 
bility P(0, n 0 ) = (1 + E)~ n ° that all of the Blue units 
will have been annihilated and none of the Red units. 
If this should have occurred, the battle would end 
then and there. There is also a possibility that the 
battle will end with one Red unit lost and all of the 
Blue units lost. The probability of this occurring is 

P(l, no). 

It is not difficult to see that the probabilities 
P(a, n 0 ) and P(ra 0 , j3) are not obtained from the 
formula (6). Further detailed analysis shows that the 
correct formulas for these special cases are: 


P(a, n Q ) 


(a + n 0 — 1) ! E a 
a\ (n 0 — 1) ! (l + P) a+no ’ 


P(m 0 , j8) 


(fl + mp - 1) ! (7) 

j8 ! (m 0 — 1) ! (1 + P/+V ; 


P(m 0 , n 0 ) = 0 . 


CONFIDENTIAL 


68 


STRATEGICAL KINEMATICS 


NO, RED UNITS LOST* a 

0 1 2 777 o~l w o 


0 

II 

1- 1 

</> 

o 

J 2 

if) 
t 
z 

3 

UJ 

3 n 0 ~T 

CD 

§ 

^(0,0) 

/’(l.o) 

[<* 


P(m 0 - 1,0) 



7 

^(1,1) 

a 



r 

P(W 0 \) | 

^ 





1 l 



\ 


$ 


/’( 0 ,n 0 " 1 > 





^(Q,y 

m,» 0 > 


f c 

^(w 0 -l,» 0 ) 



Table 1 


All these probabilities can be expressed in a tabular 
form as shown in Table 1. The heavy dashed lines a, 
b, and c correspond to the situation at different 
“times” T. For instance, for the line a, T = 2; for the 
line b, T = n 0 + 1 . In case b, the cells crossed by the 
horizontal and vertical portions of the dashed line 
represent finished battles; those crossed by the diag- 
onal portion of the line represent battles not yet 
finished. Line c represents T > m 0 + n 0 after all pos- 
sible battles have finished. The sum of all the P’s 
along any one of the dashed lines equals unity (as, of 
course, they must) . 

It will be apparent that for times corresponding to 
the lines b or c, the average number of combatants 
lost will not correspond to equations (1) for the Lan- 


chester law. In particular, a study of the case repre- 
sented by line c shows that when the battle is con- 
tinued to its finish, the result will be either a number 
of Blue units left or a number of Red units left. For 
any particular values of ra 0 or n 0 or E, the average 
number of survivors can be computed for these alter- 
native possibilities. 

As an example, a table of the form shown in Table 
1 can be computed for the case where there are ini- 
tially five Red units and three Blue units, and where 
the exchange rate is unity. From this tabular form 
one can compute the average number of combatants 
surviving on each side after T combats. The results 
of these calculations are tabulated under “Prob.” 
in the following: 


Table 2 


tyiq — 5; 7io — 3 ; E — 1 . 


T 

0 

1 

2 

3 

4 

5 

6 

7 + 

Avg m, Prob. 

5.0 

4.5 

4.0 

3.5 

3.06 

2.72 

2.48 

2.367 

Avg m, Lan. 

5.0 

4.5 

4.0 

3.5 

3.0 

2.5 

2.0 

2.0 

Avg n, Prob. 

3.0 

2.5 

2.0 

1.5 

1.06 

0.72 

0.48 

0.367 

Avg n, Lan. 

3.0 

2.5 

2.0 

1.5 

1.0 

0.5 

0 

0 


Prob. that the Red forces are annihilated = (29/128) = 0.2266 > 

7/ the Blues win, the expected no. of Blue survivors = 1.621 
Prob. that the Blue forces are annihilated = (99/128) = 0.7734 
7/ the Reds win, the expected no. of Red survivors = 3.061 


CONFIDENTIAL 


PROBABILITY ANALYSIS OF LANCHESTER’S EQUATIONS 


69 


The results of the probability calculation are com- 
pared with the solutions of Lanchester’s equations 
(labeled “Lan.”). We see that for this case there is 
approximately one chance in four that the battle will 
end with all the Red forces annihilated, and on the 
average, approximately 1.6 Blue units left. In the 
other three cases out of four (approximately) the Blue 
forces will be annihilated and an average of three Red 
units will be left. The limitations of Lanchester’s 
equations in the latter part of the battle are obvious. 
Of course, it should be pointed out that for larger 
numbers the per cent deviations from Lanchester’s 
laws would be smaller. 


4.3.2 The Square Law 


In order to make a probability analysis of the Lan- 
chester square law, we shall have to define an engage- 
ment as being an exchange of salvos, or a single 
attack of short enough duration so that the losses on 
each side cause no appreciable diminution in fire 
power during the engagement. Suppose that at the 
beginning of the engagement there are m 0 Red units 
and n 0 Blue units. Suppose also that during the en- 
gagement each Red unit shoots a certain fraction of 
the Blue units and vice versa. To correspond with 
the notation of equation (5), we should have the 
fraction of the Blue units shot by a Red unit, on the 
average, as ( r\/E/n ), and the fraction of the Red 
units which are shot by the Blue unit, on the average, 
as ( r/m\/E ), where r is the duration of the engage- 
ment in the new units of time, defined by equation 
(4). 


There will be a certain number of units which are 
hit more than once. We are, however, interested in 
those units on each side which are not hit after the 
engagement. The probability that a given Red unit 
is not hit is given by the expression [ 1 — (r/my/E) ] n . 
Again using the multinomial distribution, we find 
that the probability that a Red units are hit out of 
the total of m Red units, and the probability that a 
number (3 of the Blue units are hit during the engage- 
ment, is given by equation (8) . 

T \n(m- a) 

m\/E ) 


Pr(a) 


m 


a \{m — a) 


-a)!V 


P„(.B) = 


B ! (n - 0) 


I—(: 

— B) l \ 


[‘- 0 - 


vi)'] 


m 

y/E\ m(n ~ a 

n ) 


( 8 ) 



From these expressions one can find the average num- 
ber of Red and Blue units hit during the engagement. 
These expressions are 

«av S = m[l-(l-— 7=)], 

(9) 



This expression does not correspond to the solution 
of Lanchester’s equations except in the limit of small 
values of r. 

If r is not particularly small, but if the number of 
combatants on both sides is quite large, the equation 
may be reduced to somewhat more simple form 


«avg = m[ 1 — e-Cnr/m^)] ? 


13 avg = n [1 — e-(»»rVl/n)] f 


( 10 ) 


which still do not correspond with the solution of 
Lanchester’s equations. If, however, the quantities 
in the exponential are quite small, as they would be 
if the engagement is considered to last a very short 
duration dr, then the number of Red units lost 
(which equals —dm) is equal to — (n/\^ E) dr, which 
checks completely with equation (5) . 

By going back to equation (8), however, we can 
extend the differential technique to the probabilities 
themselves. For instance, if we define P(ra, n, t) as 
the chance that at the time t there are m Red units 
and n Blue units still unhit, then a detailed study of 
the elementary engagement lasting a time dt shows 
that the probability functions satisfy the following 
recursion relations : 


— P(m,n,t ) = (m\/ r E)[P(m, n+1, t)— P{m, n,t)] 
dt 

+ (^ 7 |) [ p( - m + 1( n ’ - p( ' m ’ n > » 

(ii) 

— P(m, 0, t) = {my/ E)P(m, 1 , t) , 
dt 


These equations can be solved subject to the initial 
condition that P(m 0 , n 0 , t) equals unity and all other 
P’s equal zero at t = 0. The calculations are tedious 
for large numbers, but are straightforward. 


CONFIDENTIAL 


70 


STRATEGICAL KINEMATICS 


4.3.3 An Example 

Detailed study of the solutions of equation (11) 
shows that the average values of m and n, as func- 
tions of time, are fairly accurately equal to those pre- 
dicted by the solution of Lanchester’s equations (5) 
for the early stages of the battle. During the later 
stages, however, deviations occur from the Lanchester 
solution of a nature analogous to those displayed in 


of the nonzero probabilities give the chances of the 
various outcomes. In the example given in Table 3, 
the probabilities of the eventual results are expressed 
in Table 4. Therefore in the long run the chances are 
about 9 to 1 that the Reds will win. If they do win, 
they will have approximately four combatants sur- 
viving. The Blues will have one chance in nine of win- 
ning, and, if they win, they will have approximately 
two combatants surviving. 


Table 3 

m 0 = 5, n 0 = 3, E = 1; expressions for P(m, n, t) 
No. Blue units remaining = n 



3 

2 

1 


5 

e -st 

5e~ 7t (1 - e~‘) 

25 

— e~ 6t (1 - e~0 2 

2 


4 

Se~ 7t (1 - e-*) 

lle _6< (1 - e~ 1 ) 2 

113 

e -5< (1 _ e ~t) 3 

6 

No. Red units 





remaining 
= m 

3 

9 

- e -6* (! _ e ~t) 2 

2 

71 

— e~ 5t (l - e~ 1 ) 3 

6 

163 

— e-«(l -e~'Y 


2 

9 

-e~ 5t (1 - e~ 1 ) 3 

49 

_ e -U (1 _ e -t) 4 

6 

359 

e~ 3t (1 -e~9 5 

60 


1 

27 

_ e -4<(l - e -<)4 

8 

473 

e-« (1 - e“9 5 

120 

1191 

e~ 2t (1 - e~ 1 ) 6 

720 


Table 2. An example will perhaps illustrate the na- 
ture of the phenomenon. We choose the same initial 
number of combatants as that chosen for Table 2, 
in order to compare the probability calculations for 
the linear and the square law cases. Table 3 gives 
values of the probability functions except for m = 0, 
or n = 0. 

The functions P(m, 0, t) and P(0, n, t) can be com- 
puted from the last two of equations (11) by simple 
integration. 

Table 3 differs from Table 1 in that the time enters 
into each expression in the present case; whereas the 
time is indicated by the dashed line in the former 
case. Here the sum of all the P’s is equal to unity at 
all times, whereas in the earlier case the sum of the 
quantities along each dashed line is equal to unity. 
In the present case the eventual result is obtained by 
letting t go to infinity. When this is done, only the 
lowest row (for m = 0) and the right-hand column 
(for n = 0) will differ from zero, indicating that the 
battle has come to an end with some of one side or 
some of the other side surviving. The limiting values 


The difference between these results and those 
given in Table 3 for the linear law are quite interest- 
ing. For the linear case the chance of the Blues sur- 
viving was one in four, approximately; whereas for 


Table 4 

m 0 = 5, n 0 = 3, E = 1; limiting values of P(ra, 0, t) and 
P(0, n , t). 


m = 5, P(m, 0, oo ) = 

0.3721 

n = 3, P(0, n, oo ) = 

0.0362 

4 

0.2690 



3 

0.1456 

2 

0.0469 

2 

0.0712 



1 

0.0295 

1 

0.0295 

Prob. Reds win = 

0.8874 

Prob. Blues win = 

0.1126 

with 3.994 survivors ex- 

with 2.059 survivors ex- 

pected if Reds win. 


pected if Blues win. 



the square law the chance of survival is one in nine, 
approximately. This corresponds to the increased im- 
portance of outnumbering the opponent in the square 
law case. The other striking difference is in the num- 


CONFIDENTIAL 


THE GENERALIZED LANCHESTER EQUATIONS 


71 


ber of survivors. In the linear case, if Red wins the 
expected number of surviving victors is three, where- 
as in the square case four are expected to remain. The 
expected numbers*, assuming a Blue victory, are 1.6 
and 2, respectively. In general, therefore, one can 
expect a larger number of surviving victors for the 
square law case, a result which again illustrates the 
advantage in numbers which the square law case rep- 
resents. Even if the weaker side is lucky, and hap- 
pens to win (it can happen in the case in question 
once in nine tries), this luck will most likely turn up 
early in the battle by a chance reversal of the nu- 
merical advantage. Once this reversal does occur, the 
Blues can overwhelm the remaining Reds without 
much additional loss, and end up by having wiped 
out the Reds with an average loss to the Blues of only 
one unit. 

We have shown by these two examples that any 
differential equations representing war conditions 
(such as Lanchester’s equations) have their limita- 
tions due to the fact that chance enters into the 
actual battle, and the exact outcome can never be 
predicted accurately. As long as the equations are 
not pressed too hard (such as by going to the limit of 
annihilation of one force), however, the solutions of 
the equations will correspond quite closely to the 
“expected value” obtained from the probability 
analysis. One must expect the actual results to de- 
viate from the expected values, with the average 
deviation increasing as the solutions tend toward the 
ultimate annihilation of one force. 

4.4 THE GENERALIZED LANCHESTER 
EQUATIONS 

The previous sections have dealt with the applica- 
tion of Lanchester’s equations to more or less con- 
tinuous engagements — to battles rather than wars. 
Sometimes it is of interest to utilize the same sort of 
analysis in discussing the overall trend of a war, 
though any attempt to reduce the course of a war to 
the scope of a set of differential equations is such a 
sweeping simplification that we should not expect the 
results often to correspond closely to reality. A dis- 
cussion of the problems involved and the nature of 
the resulting solutions is, however, of considerable 
interest, if only as a basis of comparison with reality. 

In the first place, there is the question of the units 
of measure of the quantities m and n, the fighting 
strengths of the opposed forces. Each side has, at any 
moment, a certain number of trained men, of battle- 


ships, planes, tanks, etc., which can be thrown into 
battle in a fairly short time, as fast as transport can 
get them to the scene of action. The total strength of 
this force is determined by the effectiveness of each 
component part (as discussed in Section 3.3). At any 
stage of war we can say, very approximately, that a 
battleship is as valuable as so many armies, that a 
submarine is as valuable as so many squadrons of 
planes, etc. To this crude approximation, each unit 
can be measured in terms of some arbitrary unit — so 
many equivalent army divisions, for instance. Na- 
turally, differences between submarines and tanks are 
qualitative as well as quantitative, and neglecting the 
qualitative side is part of the oversimplification of the 
present analysis. When the forces involved are large, 
however, the quantitative aspect begins to oversha- 
dow the qualitative, and we can begin to think of a 
number which is the measure of the total fighting 
strength of a nation at some instant. 

This strength is continually changing with time. 
In the first place, both sides are busy producing more 
strength: training men, building planes, etc. The rate 
of production (in the same units as m or n are meas- 
ured) for the Red side is P, and that for the Blue side 
is Q. These will vary with time, but for our first 
analysis we will assume they are constant. 

4.4.1 Loss Rates 

In addition, the strengths will be decreasing due to 
the fighting. This rate of destruction must depend on 
the strengths of the two sides, and it is not certain 
what form of function most nearly represents the 
behavior of actual wars. Certainly a Lanchester term 
of the form (—an) for the rate of change of m is a 
reasonable one, since the rate of loss of Red units 
must increase as the Blue strength increases. But 
there is also a term proportional to m needed in the rate 
of loss of m, to represent operational attrition. The 
resulting expression for the Red operational loss rate, 
(—an— cm), is the simplest expression which can rep- 
resent the overall behavior of war. When the Red 
strength is considerably larger than the Blue strength, 
then the Red side will determine the rate of fighting, 
and will work out replacement schedules for forces in 
action, so that it is not unreasonable to expect that 
the percentage losses for the Reds will be constant 
(i.e., the Red loss rate will be proportional to m) and 
that the Blue loss rate will also be proportional to m. 
If the opposed strengths are about equal, we would 
expect that the loss rates of both sides would be 


CONFIDENTIAL 


72 


STRATEGICAL KINEMATICS 


proportional to the opposed strengths. Consequently, 
only linear terms in m and n should be included. 
These requirements are all met by the expression 
{—an — cm). 

The generalized Lanchester equations are therefore 



an — cm , 


— = Q — bm — dn , (12) 

dt 


increases, whereas the exponential in the third term 
continually diminishes. Consequently, if the con- 
stant E, as determined by the initial conditions, turns 
out to be positive, the Blue forces (n) are eventually 
annihilated, and, if E is negative, the Red forces (m) 
eventually go to zero. 

When the opposed units are equally effective, 
a = b and c = d. In this case the equations take on a 
simpler form: 

a = b = fx] c = d = X; k = 0; 


where, in general, a and b are larger than c or d. At 
first, we will consider the production rates, P and Q, 
to be constant. Differential equations 13 of this sort 
have been discussed in relation to the struggle be- 
tween animal species 11 - 12 and in chemical kinetics. 
The unit of time can be the year. The quantity P will 
then be the number of equivalent armies (or the 
equivalent number of battleships) which the Red 
nation can train and equip in a year, and so on. 

The solutions for equations (12) are : 

m = A + Ee ill - X)t + Fe _( '‘ +X) ' . 

b 

n = B - Ee^-^ 1 + Fe _( '‘ +X)f ; 

a 


X = ^{c + d ) ; k = ^{c — d); fi = a/k 2 + ab] 


A _ Qa—Pd 
ab—cd 


Pb-Qc 

r ; ab — cd = n 2 — \ 2 ; 

ab — cd ’ 


m= A+ Ee {a ~ c)t + Fe~ {a+C)t ; 
n = B - Ee (a ~ c)t + Fe~ {a+c)t ; 


A 


Qa — Pc 
a 2 — c 2 


} 


Pa — Qc 
a 2 — c 2 ’ 


(14) 


E 

F 


1 

2 

1 

2 





P 

a -T c 



n 0 — 


Q 

a + c 



Here again the sign of the constant E, fixed by the 
initial conditions, determines which of the forces goes 
to zero. Examining this factor, we see that the total 
strength of one side is equal to its initial fighting 
strength, plus the productive rate divided by the 
quantity (a — c) . The sign of E depends on which of 
these strengths is the largest. 


E = 


ab 


2m(m + 




m 0 + <L±Ji±K p 


ab — cd 


(13) 


M + 

b 


F = 


ab 


2 m(m + k) 


-■[ 

M + 

~L 


i c n — k JT[ 

rk+ ^r^- Q \) ; 


+ Wo + 


ab 


~ » + K pi 
h — cd J 

^q] 

— cd J 


Since ab is larger than cd, in general, we have /z 
greater than X. Therefore, the exponential in the 
second term of the equations for m and n continually 

b The biological equations contain a nonlinear term, pro- 
portional to the product (nw), which does not seem to be justi- 
fied in the case considered in this volume. 


4.4.2 Typical Solutions 

As an example of the behavior of the solutions of 
Lanchester’s generalized equations, Figure 1 shows 
the results for eight different cases of initial strength. 
The attrition rates have been chosen as follows: 
a — 2, c = 1. The top set of four curves corresponds 
to the case when the initial fighting strengths of the 
opposed forces are equal; the lower four curves cor- 
respond to the case where the initial fighting strength 
of the Red forces is twice that of the Blue forces. 
The first two curves on the top row present cases 
where initial forces and productive strengths are 
equally matched, so that the battle ends in a draw. 
In the other cases, either the initial forces, the pro- 
ductive strengths, or both differ, so that one side or 
the other is eventually wiped out. 

The last curve on the bottom row is of particular 
interest. It represents a case where the eventual 


CONFIDENTIAL 


THE GENERALIZED LANCHESTER EQUATIONS 


73 


</> 

Ld 

O 

£T 

O 

Li. 


TIME 



Figure 1. Solutions of the generalized Lanchester equations for a = b = 2, c = d = 1; for different productive capac- 
ities P and Q and different initial forces m 0 and no. ( dm/dt ) = P — an — cm; (dn/dt) = Q = bm — dn. 


winner started out with a two to one handicap in 
initial fighting strength. This initial disadvantage 
was more than overcome by a three to one produc- 
tion advantage. For the first third of the conflict, the 
Blue forces were still further depleted, and for more 
than half of the duration of the conflict the Blue 
forces were outnumbered by the Red forces. Once the 
initial handicap had been overcome by the larger 
production, however, the advantage rapidly became 
decisive and the Red forces were wiped out in short 
order. The increasing rapidity of the final debacle 
is a characteristic of Lanchester’s equations and is 
met with, to a certain extent, in actual warfare. 


minishes as the enemy’s strategic forces increase, and 
increases as its own defensive forces increase. Each 
side must apportion its forces between defense and 
strategic offense, so as simultaneously to diminish the 
enemy’s productive capacity and to wipe out the 
enemy’s defensive forces in the most expeditious 
manner. 

It will be of interest to work out a crude approxi- 
mation to this state of affairs. We assume that both 
sides divide their total forces into two parts: 

m = m t + m s ; n = n t + n 8 ; 

m t , n t : tactical forces; m 8 , n s , : strategic forces. 


4 -4.3 Destruction of Production 

The generalized Lanchester’s equations, discussed 
above, are capable of somewhat more sophisticated 
interpretation than has been given them in the first 
part of this section. We there assumed that each 
combatant maintained his productive capacity con- 
stant. This never has been exactly true, and since the 
advent of the strategic airforce it is far from being 
true. The productive capacity of a nation depends on 
the strength of the enemy’s strategic forces, and also 
on the strength of its own defenses. Production di- 


The strategic forces are directed only against the 
enemy’s productive capacity, whereas the tactical 
forces are directed against the enemy’s strategic and 
tactical forces. The tactical forces are the ‘‘fighting 
forces” and their attrition rates, therefore, corre- 
spond to the generalized Lanchester’s equation dis- 
cussed above (at least to the rough degree of 
approximation which concerns us here) . 

The effect of the strategic forces is shown in a 
modification of the enemy’s productive rate. It takes 
a certain amount of strategic force to keep a certain 
amount of the enemy’s factories out of commission. 


CONFIDENTIAL 


74 


STRATEGICAL KINEMATICS 


Therefore, one might expect, to the first approxima- 
tion, that the diminution in the enemy’s production 
is proportional to the strength of one’s strategic 
force. The effectiveness of this force, however, de- 
pends on the strength of the enemy’s tactical force, 
which in part defends his productive capacity. To a 
very crude approximation, one might expect that 
this factor of effectiveness, for diminution of the 
enemy’s productive capacity, would be proportional 
to the ratio between the strategic force and the oppos- 
ing tactical force. In other words, the simplest pos- 
sible formula representing the expected behavior is 
as follows: 

Red production = P ( 1 — /5 n s ) ; 

\ m t J 

Blue production — Q [l — p — ra s ) . 

V n t / 


4.4.4 Tactical and Strategic Forces 

These equations cannot be solved immediately be- 
cause we have not as yet laid down any rules as to the 
relative strengths of tactical and strategic forces. The 
commanding generals of the two sides must decide 
these distributions. Their decisions, of course, will be 
based on a great many things : politics, details of pro- 
duction, efficiency of intelligence service, etc. Pre- 
sumably each side should distribute its forces be- 
tween the tactical and strategic arms, in such a way 
as to make one’s own loss rate as small as possible, 
and one’s enemy’s loss rate as large as possible. In 
terms of the crude model we are here considering, the 
commanding general of the Red side should strive 
to make the expressions : 

d^n_dn = p_Q_Sp (n - Q 2 _ ^ (in - m t ) 2 ~ I 

dt dt L m t n t J 


The formulas lead to absurdities if the ratio be- 
tween strategic and tactical forces become too large. 
Nevertheless, it is not difficult to see that, within 
reasonable limits, these formulas are crude approxi- 
mations to the behavior we have been discussing. We 
have made a further simplifying assumption in mak- 
ing the coefficients in the parentheses both equal to /3. 
Our only excuse for thus limiting ourselves to cases 
where the opposing strategic forces are equally effec- 
tive is that this assumption is not unreasonable, and 
any further complications introduced now will render 
the final solution too complicated for easy under- 
standing. The more complicated case can be worked 
out by the reader, if this is desirable. 

Our equations for the increase and decrease of the 
opposing forces, therefore, turn out to be: 


~ = P~Pl S^-afa + md; 
dt m t 

~ = Q - Q/3— - a(m t -f- n t ) . 
dt n t 


(15) 


Here we have again simplified matters by letting the 
coefficients of the Lanchester terms all be equal to 
the same quantity, a. Our justification is again that 
this simplification does not invalidate the general be- 
havior of the solutions, and it simplifies the formulas 
considerably. Once the behavior of the simplified 
equation is discussed, further complications can be 
added as desired. 


= L(m t , n t ) 


as large as possible; and the commanding general of 
the Blue side should strive to make this same quan- 
tity as small as possible. At each instant, the values 
of (m, n) are fixed by the previous history of the 
situation. The Red general, at each instant, must 
adjust m t so that the quantity L is as large as possible. 

This is an example of the “minimax principle” 
which is discussed in more detail in Chapter 5. In 
actual practice, each general must make his decision 
on inadequate knowledge of what the other general 
has decided. The “safest” decision for each general is 
to assume that the other general has made the best 
possible choice (for his side). This means that the 
Red general must assume that the Blue general is 
trying to minimize L, and the Blue general must 
assume that the Red general is trying to maximize L. 
These simultaneous adjustments can be made by 
requiring : 


dL d 2 L dL d 2 L 

— = 0 ; <0; — = 0; — > o • (16) 

dm t dm? dn t dn? * ’ 1 ; 


which is the minimax principle in one of its mathe- 
matical forms. If the Red general makes his choice 
according to these equations, then his situation will 
be as good as possible if the Blue general makes the 
corresponding choice for n t . If the Blue general does not 
make this choice for the relative distribution of 
strength between tactical and strategic forces, then 


CONFIDENTIAL 


THE GENERALIZED LANCHESTER EQUATIONS 


75 


the Red general can always improve his situation by 
appropriate modification of his balance between tac- 
tical and strategic forces. Consequently, the distribu- 
tion above forms the safest solution of the problem 
with the forces at hand, and will be called the “basic 
solution.” It is the best solution possible when the 
two opponents have equal intelligence; if one side 
departs from this solution, the other side can obtain 
still better results. 


4.4.5 The Minimax Principle 

Applying the minimax principle to the approxi- 
mate expression for L, we find that : 


P • n t {n — n t ) 2 = 2 Q • m t 2 (m — m t ) ; 


2 Pn t 2 (n — n t ) = Qm t (m — ra<) 2 ; 


m t = - (n — n t ) ; n t = — (m — m t ) ; 

2 2 p 


(17) 


where p 3 = ( P/Q ). Therefore 


‘ = (l pn -\ m )’ nt= ^\ n -r)’ 

8 = ^(2 m — pn) = 2pn t ;n s = ~ (^2n — — ^ 


is the basic solution, as long as m < 2pn and n<2m/p. 

These solutions are very interesting. They show 
that, within certain limits, the size of the tactical 
force of one side should increase if the enemy’s total 
forces increase (i.e., the fraction of Red forces which 
should be assigned to the tactical arm depends 
linearly on the ratio between the Red and the Blue 
total forces). It also depends on the ratio between 
the initial productive forces of the two sides, through 
the quantity p, although the dependence on this 
ratio is only to the one-third power. We notice that 
these formulas would require the value of m t to be 
sometimes negative, when the ratios of the two forces 
become considerably unbalanced. This is of course 
due to the crudity of our initial equations, and the 
solutions will have to be watched to prevent such 
absurdities from arising. Aside from these crudities, 
the solution does correspond to what we might ex- 
pect. If the enemy strength increases, we put more 
of our forces in the tactical arm. If our production is 
large, we need a somewhat greater defensive strength 


VALUES OF RED PRODUCTION RATE 



TOTAL RED LOSS RATE DUE TO FIGHTING 



Figure 2. Contour plots of assumed production rates 
and total losses as function of own tactical force and 
enemy tactical and strategic force. 


(tactical force). On the other hand, if our own fight- 
ing forces are larger than the enemy’s, we can afford 
to put more of our strength in the strategic arm, and 
so on. 

If we now assume that the generals on both sides 
continually adjust their forces, so as to correspond 
to the “basic solution,” then it turns out that 
equations (17) inserted in equations (15) correspond 


CONFIDENTIAL 


76 


STRATEGICAL KINEMATICS 



ITJ ^ 

m = n s 1 m = 1.5, n=1 m= 2 , n=1 

Figure 3. Contour plots of Red and Blue total loss rate, and of differential loss rate, for different values of total forces, 
for production and losses as given in Figure 2. Points marked M on the differential loss rate plots are the minimax points, 
corresponding to safe operation. 


to the generalized Lanchester’s equations (12), with 
the following constants: 


a = + 2 P- l] 

, 5 ,.r8»rf + 2_ i 

3 L pa J 

3L a p J 


( 18 ) 

c = b - 4 QpP- d = a- 4 ; p = 

A graphical presentation of these arguments will 
perhaps make this more clear. In so doing, we can 
use somewhat less crude expressions for production 
rate and for loss rate due to fighting. Figure 2 shows 
possible contour plots for these quantities as func- 
tions of the strategic and tactical forces. The upper 
contour is for the Red production rate. If the Blue 
strategic force n s is small and the Red tactical force 
m t is large, the Red production rate has its full value. 
If the relative strengths are reversed, the production 
rate falls nearly to zero. The lower contour is the total 


Red loss rate due to fighting. This loss rate is zero if 
both tactical forces are zero, and it increases linearly 
with increase in either force, according to the general 
Lanchester term. The Red loss rate depends more 
strongly on the size of the Blue tactical force, n t than 
on m t , as mentioned earlier. 

These curves can be combined in various ways to 
obtain the Red or Blue net gain as a function of (m t , 
n t ) for various values of m and n. This has been done 
for several different relative total strengths in Figure 
3. The upper three sets of contours display values of 
the net rate of increase for Red forces, and the middle 
row shows values of the corresponding increase for 
the Blue forces. Negative values mean net loss rate, 
and positive values mean net gain per unit of time. 
The bottom row of contours shows values of the 
function L, the difference between Red and Blue 
gain. According to equation (16) a minimax point is 
to be found on these surfaces. 

The minimax points are marked on the contours 


CONFIDENTIAL 


REACTION RATE PROBLEMS 


77 


by M . Examining the center plot of the bottom row, 
for m = 1.5, n = 1.0, we see that the minimax point 
corresponds approximately to m t = 0.65 and n t = 

0.55. If the Red general changes his relative distribu- 
tion of tactical and strategic forces, making n t equal 
to 1.0, for instance, then the Blue general, by corre- 
spondingly increasing the Blue tactical forces, can 
reduce the Blue net loss and increase the Red net loss. 
Consequently, it is safest for the Red general to dis- 
tribute his forces corresponding to the minimax point, 
at least until he can determine whether the Blue 
general is doing likewise. If the Blue general has not 
done so, then the Red general can adjust m t to im- 
prove the situation, as can be seen from the contour. 

The differential equations corresponding to these 
loss rates can be solved numerically. A contour plot 
for L has to be drawn for each instant of the war; the 
proper distribution of tactical and strategic forces 
can then be determined, and the corresponding loss 
rates for the two sides can be computed. This is then 
inserted back into the equations for the rate of change 
of ( m , n ) to obtain a final solution. 

4.5 REACTION RATE PROBLEMS 

Many problems concerned with the increase and 
attrition of forces can be analyzed by equations 
closely related to those used in chemistry to study 
reaction rates. An example of this can be taken from a 
partial analysis of the antisubmarine war in the At- 
lantic. The study concerns itself with the general 
problem of air offensive action against the German 
submarine. It assumes that a certain amount of air- 
craft and number of crews are available for action 
against the submarine, over and above the number 
of aircraft and crews needed for protection of con- 
voys. It discusses the question as to how this offen- 
sive action should be distributed in order most effec- 
tively to reduce the total number of submarines in 
the Atlantic at any time. 

Three distinct types of offensive action against 
enemy submarines can be taken. 

1. Submarines can be hunted out in the Atlantic, 
and can be sunk or damaged there. 

2. The submarine repair bases along the coast of 
France can be bombed, so that fewer submarines per 
month can be reserviced and put to sea again. 

3. The factories in Germany which produce sub- 
marines can be bombed, so as to reduce the produc- 
tion rate. 

Each of these offensive actions has its effect in re- 


ducing the number of submarines in the Atlantic, 
either immediately or at some future date. An im- 
portant strategic question, which must be decided 
from time to time, is how the available offensive air 
strength is to be distributed among these three ac- 
tivities in order to produce the greatest reduction in 
submarines in the Atlantic at the time when it is most 
needed. Before deciding on the relative apportion- 
ment of strength, a great number of different factors 
must be taken into consideration. Along with other 
factors, it is possible that a purely theoretical study 
of the effects on the submarine distribution of 
changes in production, sinking, or repair may be 
worth consideration. It is certain that the analysis 
summarized in the following pages is entirely too sim- 
plified to represent the actual case in all its complica- 
tions. Nevertheless, it is felt that the results of this 
simple theoretical analysis should prove suggestive 
as to actual possibilities. 

4.5.1 Circulation of U-Boats 

In order to study theoretically the relative effect of 
damaging the factories or the repair bases, or in at- 
tacking the submarine directly in the Atlantic, we 
must study the activities of the average submarine. 
Submarines are produced at an average rate P per 
month, and are being sunk at an average rate S per 
month. Therefore, the net increase in their number 
per month is P — S which is called /.We know that 
the average length of time the submarine stayed on 
patrol on the Atlantic was about two months. After 
this time the submarine went back to one of the bases 
along the French coast for repair, refueling, and rest- 
ing the crew. Therefore, on the average, about half 
the number of submarines in the Atlantic returned 
to their base each month. 

The length of time the submarine remained at the 
repair base depended on the amount of repair work 
which had to be carried out, and on the degree of 
efficiency of the base itself. The average amount of 
repair work required depended on the average num- 
ber of U-boats which were damaged each month, and 
the efficiency of the base depended on the amount of 
damage the base had received that month. The de- 
tails of this interrelation will be discussed more fully 
later on. At this point it is only necessary to notice 
that the repair bases had a maximum capacity for 
refitting submarines, which capacity could be dimin- 
ished either by bombing the bases or by increasing 
the average amount of damage to a submarine on 


CONFIDENTIAL 


78 


STRATEGICAL KINEMATICS 


patrol. Thus the return flow of submarines from the 
repair bases to the Atlantic constituted a bottleneck, 
whose size had an important effect on the total num- 
ber of submarines in the Atlantic at any one time. 

In fact, it is possible to see that a marked reduc- 
tion in the flow rate L of submarines from bases to 
Atlantic would produce an effect on the number of 
submarines in the Atlantic in a relatively short time. 
This is due to the fact that the U-boats remained in 
the Atlantic no longer than about two months, so 
that after a period of two months all the submarines 
which were originally in the Atlantic had been re- 
placed by submarines which came from the repair 
bases within the two months time. 

In order to obtain results of a more quantitative 
nature, we must make certain reasonable assump- 
tions about the interrelationships between the var- 
ious rates and numbers : 

We define the following quantities: 

A = average number of U /B in Atlantic ; 

B = average number of U /B in bases ; 

P = production of U/B per month; con- 
stant; 

S = number of U/B sunk per month; 

P — S = I = net increase in number of U /B per 
month; 

t = time in months; 

L = number U/B leaving bases per month; 
L = M(l - e~ CB/M ); 

M = maximum rate of repairing U/B and re- 
turning them to Atlantic ; 

CB = rate of repairing U/B in very lightly 
filled base; C is usually 1 ; 

1/K = mean length of stay of U/B in Atlan- 
tic — 2 months. 

The equation for L represents our assumption con- 
cerning the capabilities of the repair bases. We 
assume that the average rate of sending submarines 
back from the bases to the Atlantic depends on the 
number in the bases at any one time, in the manner 
given by the equation. This indicates that if there are 
a small number of submarines in the bases then the 
repair work can proceed efficiently enough so that 
the submarines can be sent out again about a month 
after they have come in from their previous cruise. 
The equation shows this; for small values of B, the 
rate of leaving, L, is approximately equal to the 
number, B, in the bases. When there are a large num- 
ber of submarines present in the bases, however, the 
state of repair of the bases and the average damage 
to the submarines begins to make itself felt. We 


assume that there is a fixed maximum rate of repair- 
ing submarines at any given time, which number is 
indicated by M on the plot. It is assumed that, at the 
time considered, no more than this number, M, can 
be put into operation each month, no matter how 
many submarines are present in the bases awaiting 
repair. The curve for L therefore never rises higher 
than the value M . 

For the purposes of this study, it is the value of M 
which indicates the state of efficiency of the base. Any 
increase in damage to the bases by bombing them, or 
any increase in average damage suffered by subma- 
rines in the Atlantic, will decrease the value of M 
temporarily. The question of the relative effect of 
damaging the factories and damaging the bases, 
therefore, resolves itself to the question of the rela- 
tive effect of a change in I = P — S and a change 
in M. 


4.5.2 Equations of Flow 

With these assumptions, the equations for the 
flow of submarines can be set up. The fundamental 
equations for the change of A and B can be written 
in dimensionless form, if the variables are changed in 
the following manner: 


CA CB 

x = — ; y = — ; 
M M 




u = Ct . 


Then the operations become : 

— = p — kx + (1 — e ~ y ) ; — = kx — (1 — e~ y ) . (19) 

du du 


Solutions of these equations, for the range 0<w< 
10, and for the values: p = 0, 1; k = 0, 1, 

were run off on the Differential Analyzer at MIT for 
the initial conditions: x 0 = 0, 1, 2; y 0 = 0, 1, 2. 
Graphs have been obtained from these solutions. 

For small values of u, the following series expan- 
sions hold : 

x = Xq + (p — kx 0 + 1 — e y o)u + • • • 
y = 2/o + (kx 0 — 1 + e y o)u + • • • . 

For large values of y (except for the cases k = 0 or 

V = 0): 


(P ~ 1) 


y-+x o + 2/0 




+ pu. 


CONFIDENTIAL 


REACTION RATE PROBLEMS 


79 


4.5.3 Typical Solutions 

A few typical solutions are given in Figure 4. 
Curves are given Tor the average number of U-boats 
in port and in the North Atlantic, for different times 
after the start and for different net production of 
U-boats per month. It will be noticed that at first 




MONTHS 


Figure 4. Typical solutions for the submarine “turn- 
around” problem. Plots of submarines in port and on pa- 
trol as functions of time, for different values of net increase 
of submarines I. Assuming: average length of cruise 2 
months; average stay in uncrowded port 1 month, maxi- 
mum rate of servicing U /B in port = M = maximum rate 
of sending U /B out on cruise to be 50 per month. 

the number of U-boats in the bases is less than the 
number of U-boats in the North Atlantic. However, 
after six months (for the value of M chosen in the 
example) the effect of the bottleneck in the repair 
bases begins to make itself felt. The increase in the 
number of U-boats in the North Atlantic is not as 
great as at first, and the excess U-boats pile up in the 
bases, since they can not get repaired fast enough. 

However, such solutions, starting with the begin- 
ning, are not of the greatest amount of interest for 
our purposes here. We are more interested in finding 
out what happens to the curve when we suddenly 
change M, the maximum rate of returning U-boats 
to the Atlantic, or when we suddenly change I, the 
net rate of production of U-boats. Such a sudden 
change would correspond to the serious attack on the 


bases or on the factories, or on a sudden increase in 
the offensive against U-boats in the Atlantic. A case 
in point is given by Figure 5, which shows A, the 
number of U-boats in the Atlantic before and after a 
single attack. In this case we have taken one of the 
curves from Figure 4 for the initial increase. If there 
had been no attack, the number A would have con- 
tinued along the dotted curve. At the end of six and 
a half months, however, we assume an attack either 
on the factories or on the repair bases. In the curve 

BEFORE ATTACK, 

NET INCREASE IN U/B, 

I = 25 PER MONTH 



Figure 5. Solutions for submarine flow, or “turn- 
around,” problem. Effect of damage to submarine pro- 
duction and to repair facilities. 


marked “Case 1” we assume a reduction of the net 
production to one-half its original value. In the curve 
“Case 2 ” we assume a net decrease of the maximum 
repair rate of the bases to one-half its initial value. 
In the cases shown here it would seem that reducing 
the effectiveness of the repair bases is slightly more 
efficacious than reducing the effectiveness of the fac- 
tories. This is not always the case, however. 

The curves of Figure 5 are still not exactly the 
ones which we need to answer our questions. Another 
set is given in Figure 6, this time plotted only for the 
months after the attack. No assumption has been 
made as to the antecedent curve, except that at the 
time of the attack there are 100 U-boats in the 
Atlantic and 50 U-boats in bases. (This was approxi- 
mately the case at one time during World War II.) 
The curves plotted give the number of U-boats in the 
Atlantic against the number of months after the 
attack, for different assumed values of /, net increase 
in submarines per month, and M, the maximum rate 
of repair of submarines at all bases. These curves 


CONFIDENTIAL 


80 


STRATEGICAL KINEMATICS 


show the effect of different reductions of effectiveness 
of the factories and of the repair bases. 

For some time during the war, the average value of 
I was between 12.5 and 25, and the average value of 

VALUES AFTER ATTACK CURVE NO. 

I M 

25 100 

25 50 

12.5 100 

12.5 50 
25 25 

0 100 

12.5 25 
0 50 

25 0 

0 25 

12.5 0 

0 12 3 4 

MONTHS AFTER ATTACK 

Figure 6. Solutions for the submarine flow problem. 
Effect of damage to submarine production and repair fa- 
cilities. Fifty submarines left in repair base after attack. 

I = net increase in submarines per month; M = maxi- 
mum rate of repair of submarines per month. 

M was between 50 and 100. Consequently we would 
expect that the number of U-boats in the Atlantic 
would have followed a curve somewhere between 
curves 2 and 3 of Figure 6 (curve 3 is more likely) if 
no attacks had been made on bases or factories. 

Curve 6 indicates that if we attacked the U-boat 
factories strongly, so as to reduce the net production 



FULL PRODUCTION 1 
FULL REPAIR 


FULL PRODUCTION 2 

GOOD REPAIR 
PARTIAL PRODUCTION 3 
FULL REPAIR 


PARTIAL PRODUCTION 4 
GOOD REPAIR 
FULL PRODUCTION 5 
POOR REPAIR 
NO PRODUCTION 6 

FULL REPAIR 

PARTIAL PRODUCTION 7 
POOR REPAIR 
NO PRODUCTION 8 

GOOD REPAIR 
FULL PRODUCTION 9 
NO REPAIR 

NO PRODUCTION 10 

POOR REPAIR 
PARTIAL PRODUCTION II 
NO REPAIR 


to zero, this still would not have greatly decreased 
the number of U-boats in the Atlantic in a short 
time. Curve 4 indicates that an attack on the bases 
which only reduced to one-half the maximum rate of 
repair would likewise not have diminished the num- 
ber of U-boats in the Atlantic to an appreciable 
extent. 

Curve 7 indicates that although the factories are 
not touched, an attack on the bases which reduces 
the maximum rate of repair to a quarter of its initial 
value would appreciably reduce the number of U- 
boats in the Atlantic in a few months. Curve 8 indi- 
cates that even though the factories are knocked out, 
it also requires a reduction of the bases to half their 
initial efficiency before there would be appreciable 
reduction in U-boats in the Atlantic within a reason- 
ably short time. 

For the short-term effects, therefore, these curves 
seem to indicate that the damaging of the repair 
bases had a greater effect than the damaging of the 
factories. These conclusions must be taken with some 
caution, however, since the solution here worked out 
is for a single attack at the beginning of the curves 
and for no change in rate of production or repair 
thereafter. A balance of the probable effects of other 
factors, however, would indicate that the actual 
curves would fall above the curves considered here. 
Therefore, if the present curves do not show a cer- 
tain type of attack to be satisfactory, it would not 
have been satisfactory in actual practice. 

Other curves can be drawn for other initial con- 
ditions. They are not very dissimilar to the set in 
Figure 6, and lead to no different results. 


CONFIDENTIAL 


Chapter 5 


TACTICAL ANALYSIS 


S ome important contributions have been made 
by operations research methods in the analysis of 
tactics. Many new situations arose in World War II, 
involving new equipment or new tactics on the part of 
the enemy, for which the correct tactical answer had 
to be found. An immediate answer had, of course, to 
be worked out by the forces in the field, but it often 
turned out that such pragmatic solutions could be 
improved upon through further study. The problem 
was usually approached by the operations research 
worker from two directions: the observational, and 
the analytical. At the onset of the new conditions the 
forces in the field would be forced to try a number of 
different tactics; if detailed data on the results of 
these trials could be obtained from the field, they 
could be studied statistically to see which tactic 
seemed most promising. 

These initial data, if they were complete enough, 
could be used to obtain approximate measures of 
effectiveness, and to obtain a general picture of the 
possible behavior of the forces involved. As soon as 
this general picture could be obtained, together with 
the approximate measures of effectiveness involved in 
the operations, it was then possible to study the oper- 
ations analytically. Knowing the physical capabilities 
of the equipment involved, optimum tactics could be 
worked out theoretically. For this theoretical work to 
be of much practical value, however, the magnitude 
of the constants involved must be determined, either 
from actual operational data, or from data obtained 
by carefully analyzed tactical tests. 

Section 5.1 will give illustrations where the correct 
tactics became reasonably obvious after a statistical 
study of the operational data. As soon as the average 
results from the different actions were computed, it 
became clear which was the best action to take in a 


particular circumstance. Later sections of the chapter 
will illustrate various methods of working out opti- 
mum tactics analytically, and will discuss some of 
the general principles which are often useful in such 
analysis. Methods of studying tactical tests to obtain 
measures of effectiveness will be discussed in Chap- 
ter 7. 


5.1 STATISTICAL SOLUTIONS 

An example of a case where operational data made 
clear the appropriate tactics comes from the problem 
of the ship maneuvering to dodge an incoming suicide 
plane. In spite of our combat air patrol and our anti- 
aircraft [AA] fire, a number of Japanese suicide 
planes survived long enough to make final dives on 
some of our naval units. As soon as it was clear that 
the plane was in a dive heading for a particular ship, 
this ship could attempt to avoid being hit by violent 
maneuvers, or could continue on a steady course and 
trust to its antiaircraft fire alone to destroy the 
enemy’s aim. It was important therefore, to find out 
whether radical ship maneuvers would spoil the aim 
of the incoming Kamikaze more than they would 
spoil the aim of the defensive antiaircraft fire. 

5.1.1 Damage Due to Suicide Planes 

In order to answer this question, accounts were 
collected of 477 cases where the enemy plane was 
obviously a suicide plane heading toward a particular 
ship. Thirty-six per cent of these planes, 172 of them, 
hit the ship they were aiming for; the others missed. 
As a result of the 172 hits, 27 ships were sunk. This 
is shown in the following table : 



Larger fleet units 

Smaller fleet units 



BB 

CA, CL 

CV 

CVE 

CVL 

DD, APD 
DM, DMS 

AP, APA 
AKA,AKN 

LSM 

LST, LSV 

Small 

craft 

All ships 

Number of attacks 

48 

44 

37 

241 

21 

49 

37 

477 

Per cent hits 

44 

41 

48 

36 

43 

22 

22 

36 


CONFIDENTIAL 


81 


82 


TACTICAL ANALYSIS 


Of the 477 attacks studied, only 365 reported in 
enough detail to be able to ascertain the behavior of 
the ship and the ultimate state of the plane (i.e., 
whether it was severely damaged or destroyed by 
antiaircraft fire, or not). These attacks were analyzed 
to determine the percentage of hits, for large and 
small ships, according to whether they were maneu- 
vering or not. 



Large 

units 

Small 

units 

Total 

Maneuvering 




Number of attacks 

36 

144 

180 

Per cent hits on ship 

22 

36 

33 

N onmaneuvering 




Number of attacks 

61 

124 

185 

Per cent hits on ship 

49 

26 

34 


5.1.2 The Effects of Maneuvering 

The results indicate that battleships, cruisers, and 
carriers should employ radical maneuvers when at- 
tacked by a suicide plane. The percentage of suicide 
hits on these ships is considerably smaller when they 
maneuver than when they do not. The table, of 
course, tells nothing about what sort of maneuvers 
should be employed, but it clearly demonstrates that 
these larger ships benefit from maneuvering radically 
in the face of a suicide attack. 

Destroyers and smaller fleet units, as well as 
auxiliaries, should not maneuver with radical turns, 
according to this table, because they receive a higher 
percentage of hits when they do than when they do 
not maneuver. The table does not indicate whether 
the smaller ships would profit from the use of slow 
turns, but it does show that they should not use a 
combination of high speed and full rudder. 

Part of the reason why large fleet units should 
maneuver and smaller ones should not apparently 
lies in the effect of radical maneuvers on AA effec- 



Large 

units 

Small 

units 

Total 

Maneuvering 

Number of attacks 

36 

144 

180 

Per cent AA hits on plane 

77 

59 

63 

N onmaneuvering 

Number of attacks 

61 

124 

185 

Per cent AA hits on plane 

74 

66 

69 


tiveness. This is shown in the preceding table, giving 
percentages of suicide attackers which are seriously 
damaged by AA fire during their dive attacks. 

The data reported in the table are not particularly 
accurate, since it depends on the judgment of the 
officer writing the action report as to whether the in- 
coming plane was seriously damaged or not by the 
AA fire. Such judgments are not always accurate, nor 
are they always clearly stated in the reports. Never- 
theless, the results seemed to show that for large 
units the AA fire is about as effective when the unit 
is maneuvering as it is when not maneuvering, 
whereas the fire from the smaller units seems to be 
less effective when the ship is maneuvering. The 
difference between 66 and 59 per cent is probably 
significant, considering the number of cases report- 
ing. The rolling and pitching of smaller craft, when 
performing radical maneuvers, probably upsets the 
stability of the gun platform sufficiently to cause 
serious AA errors, whereas this does not seem to be 
true in the case of larger ships. 

Dividing these data still further, into cases where 
the suicide plane came in on high dives, and other 
cases where it came in on low dives, does not seem to 
alter the conclusions concerning maneuvering. It is 
apparent from the details that, no matter what the 
dive angle, destroyers and smaller fleet units should 
not employ radical maneuvers in order to escape 
suicide bombers. 

5.1.3 The Effect of Angle of Approach 

The first three tables showed that radical maneu- 
vers were good or bad depending on the type of ship 
being considered. Nothing was said, however, about 
what maneuvers were particularly good or bad. By 
considering the effect of the suicide plane’s angle of 
approach, some notion may be had as to what, if 
any, maneuvers should be employed by the vessel 
under attack. A breakdown of the data to show this 
effect is given in the table on page 83. 

Because of the difficulties of determining angle of 
approach on maneuvering ships and because of the 
effect of maneuvers on AA effectiveness, only non- 
maneuvering ships have been considered here. Fur- 
thermore, because of the small number of attacks in 
which the required data are known, no attempt has 
been made to break the data down by ship types. 
Grouping all ships together for this study is not 
unreasonable because all ships are of the same general 
shape and the relative distribution of fire power 


CONFIDENTIAL 


STATISTICAL SOLUTIONS 


83 


around all ship types is very similar. In other words, 
there does not appear to be any reason to suppose 
that the effect of angle of approach would be mark- 
edly different among ship types. 



Per cent hits 

Number of 


on ships 

cases 

High dives 

Ahead 

100 

1 

Bow 

50 

6 

Beam 

20 

10 

Quarter 

38 

13 

Astern 

80 

5 

Low dives 

Ahead 

36 

11 

Bow 

41 

17 

Beam 

57 

23 

Quarter 

23 

13 

Astern 

39 

23 


Two facts are apparent from the table. High 
divers achieve a greater measure of success if they 
approach from an angle other than the beam, but 
low divers do best if they approach on the ship’s 
beam. Put conversely, a ship is safer if it presents 
its beam to a high diver and turns its beam away 
from a low diver. The latter fact is contrary to much 
opinion on the subject, and certainly calls for some 
explanation. 

514 Reasons for the Results 

A discussion of the relative safety of ships against 
various angles of approach must be based on two in- 
dependent arguments, that which considers the 
amount of AA fire power which can be brought to 
bear at a given angle, and that which considers the 
relative target dimensions presented to a plane ap- 
proaching from that same angle. It is the relative 
weight of these two arguments, rather than the con- 
clusion of either by itself, which must decide the issue. 

The argument concerning AA fire power is clear- 
cut. More AA fire power can be brought to bear on 
the beam than on the bow or stern. And this is true 
no matter what dive angle is being considered. Thus, 
on the basis of this argument alone, it would appear 
as though the ship were always safest if the plane 
approached from the beam, regardless of the dive 
angle. 

The argument concerning target dimensions is 
somewhat more involved. First we must consider the 
relative size of range and deflection errors made by 
suicide divers. In the case of high dives, when all 


suicide misses of 500 yards or more are eliminated, the 
average errors in the point of crash are about 50 yards 
in range and 15 yards in deflection. The range error is 
measured along the plane’s track and the deflection 
error normal to the plane’s track, assuming the 
bridge structure of the ship to be the point of aim. 
These figures are necessarily rough because of the 
lack of precision in the action reports. They are suffi- 
ciently accurate, however, to indicate that range 
errors are about three times as large as deflection 
errors. In order to take advantage of this error dis- 
tribution, it is apparent that the small dimension of 
the ship should be placed parallel to the track of a 
high diver in order to increase the safety of the ship; 
in other words, a high-diving plane should be placed 
on the beam. Thus both this argument and that con- 
cerning AA fire power indicate that the ship is safest 
if a high-diving plane; approaches from the beam. 

In the case of a low-diving plane, the problem is 
somewhat different. If the deflection error is small 
enough, and if the plane is flying only a few feet 
above the water, it is apparent that range errors are 
of little importance. The plane simply continues 
flying until it hits the ship. Put differently, a very 
large effective target in range is presented to the low- 
flying plane no matter from what angle it approaches. 
Since range errors cannot very well be taken ad- 
vantage of in this case, it will be better to take ad- 
vantage of deflection errors by placing the small di- 
mension of the ship normal to the plane’s course, or, 
in other words, by turning the beam away from the 
plane. For low divers, then, the AA fire power con- 
sideration argues that the beam is a safe aspect to 
present to the attacker, but the consideration of 
target dimensions argues that the beam is a dan- 
gerous aspect to present. The figures of the preceding 
table indicate that the second argument is the more 
important. Apparently the distribution of fire power 
around the ship does not vary sufficiently to over- 
come the differences in target dimensions presented 
to a low diver. 

Further confirmation of these results is given by 
an independent analysis of data concerning maneu- 
vering destroyers. The table on page 84 presents the 
results broken down according to dive angle and 
whether the destroyer was turning its beam toward 
or away from the plane. 

The figures clearly indicate that a maneuvering de- 
stroyer should present its beam to a high diver. They 
also indicate, but less conclusively, that the destroyer 
should not attempt to present its beam to a low 


CONFIDENTIAL 


84 


TACTICAL ANALYSIS 



Suicide 

Number 


success; 

of 


per cent 

cases 

High dives 

Maneuvering to present beam 

17 

6 

Maneuvering to turn beam away 

73 

11 

Low dives 

Maneuvering to present beam 

67 

9 

Maneuvering to turn beam away 

45 

11 


diver. Although the number of cases here is small, the 
figures do confirm the results of the analysis of angle 
of approach on nonmaneuvering ships, and hence are 
given added significance. 

5.1.5 Suggested Tactics 

On the basis of data included in this study, the 
following conclusions are justified: 

1. All ships should attempt to present their beams 
to high-diving planes and to turn their beams away 
from low-diving planes. This recommendation, it 
should be noted, is based on the assumption that no 
great difference exists in the damage done by planes 
crashing from different angles of approach. If there 
is considerable difference, it might be necessary to 
change this recommendation. 

2. Battleships, cruisers, and carriers should employ 
radical changes of course in order to evade suicide 
planes. 

3. Destroyers and smaller fleet units and all auxilia- 
ries should turn slowly to present the proper aspect 
to the diving plane, but should not turn rapidly 
enough to affect the accuracy of their AA. 

The importance of ships employing these optimum 
tactics is illustrated by the fact that only 29 per cent 
of the dives on ships using the proper tactics, as de- 
fined above, were successful, whereas 47 per cent of 
the dives were successful on ships using other than 
these tactics. 

5-1.6 Submarine Casualties 

An extremely interesting attack on a very difficult 
problem by the use of statistical analysis was the 
investigation of the causes of the losses of our own 
submarines in the Pacific during World War II. 
Except for intelligence sources, the calculations here 
have to be made by indirect methods. One hears the 
stories of those submarines which have been dam- 
aged, but have managed to return. One does not hear 
what has happened to those submarines which do not 


return. On the other hand, it is extremely important 
for the submarine forces to know what tactic of the 
enemy is causing the greatest number of these 
casualties. 

One might expect that the answer could be ob- 
tained by collecting information on the causes of 
damage for those submarines getting home, and 
extrapolating the results to the case of those which 
did not. Such an extrapolation was made for U. S. 
submarines in the Pacific; the results did not reason- 
ably explain the known losses. An extrapolation, 
using reasonable ratios of casualties to damage for 
surface and air attacks resulted in figures for ex- 
pected casualties which were only about one-third 
of the actual casualties. 

This discrepancy might, of course, have indicated 
that our expected ratio of casualties to damage was 
too small by a factor of 3; nevertheless, there was a 
fair possibility that another cause of casualty was 
entering, which did not enter into the cause of dam- 
age. If some type of enemy tactic resulted either in a 
complete miss or a total casualty, then the subma- 
rines which came back damaged could tell us nothing 
concerning this type of tactic. It was suggested that 
the effects of enemy submarines would answer this 
description; any torpedo hit would presumably so 
damage our submarine as to prevent it from return- 
ing to base, whereas a torpedo miss might be noticed 
but would not cause damage, and so might not be 
stressed in the action report. The following analysis 
was made in an attempt to estimate how many of our 
submarines could have been sunk by enemy sub- 
marines, by counting how many of our submarines 
sank Japanese submarines. 

In general, available information concerning en- 
counters of our submarines with the enemy does not 
provide answers concerning our losses. As a trite 
example, it would be illogical to attempt to estimate 
our submarine casualties caused by destroyers on the 
basis of the destroyers our submarines have sunk. 
Similarly the ability of submarines to shoot down 
enemy planes bears no relation to the ability of 
enemy planes to sink submarines. This is simply 
because for these cases there exist no common bases 
for comparison. 

5.1.7 Comparison with Japanese Submarine 
Casualties 

However, in the special case where our submarines 
encounter enemy submarines, a basis for comparison 
does exist. Although it is not true that U. S. and 


CONFIDENTIAL 


STATISTICAL SOLUTIONS 


85 


Japanese submarines are identical either in design or 
tactical use, certainly no U. S. submarine will ever 
encounter any Japanese craft more like itself than a 
Jap submarine; although opposing submersibles, 
when compared, show detailed differences, they still 
are fundamentally alike in that they operate in the 
same medium, with the same weapons, and enjoy or 
suffer the same general advantages or disadvantages. 
It is this feature of submarines vs submarines which 
permits comparative results to be deduced. 

Since direct evidence from sunk U. S. submarines 
was unobtainable, indirect evidence Avas substituted. 
The next best information to that directly concerning 
submarines lost is information concerning those 
which were attacked but missed. Presumably the 
number of U. S. submarines claiming to have been 
fired at and missed by enemy submarines bears a 
direct ratio to the number fired at and not missed, 
and consequently unable to report the action. Thus, 
for cases where our submarines were unsuccessfully 
attacked by submarine torpedoes, there must exist a 
proportionate number of attacks on our submarines 
Avith less fortunate results. 

While there are several independent methods for 
estimating our submarine losses, preference must be 
attached to those depending on the fewest assump- 
tions. The most direct approach is simply to assume 
that Japanese attacks on submarines suffer about the 
same percentage of misses that ours do, and apply 
this figure to the number of times we have been 
attacked and missed. There was no reason to suppose 
that the percentage success of attacks Japanese sub- 
marines made against U. S. submarines was greatly 
different from the percentage success of our attacks 
on Japanese submarines. 

5.1.8 The Operational Data 

From the beginning of the war to 15 June 1944 
there were 27 submarine attacks on our submarines 
and 43 attacks by our submarines on Japanese sub- 
marines. In 17 of these 43 attacks by U. S. subma- 
rines the Japanese submarine was sunk or damaged, 
and in the other 26 cases it was missed. The break- 
down by years is shown in Table 1 . 

From these data the best overall figure for our per 
cent misses is 26/43, or 60 per cent, for all types of 
attacks on Japanese submarines, whether by day or 
night, and whether surfaced or submerged; hence, we 
hit about two-thirds as many Japanese submarines 


as we missed. Applying this factor to the numbers of 
cases in which a U. S. submarine was attacked and 
missed gives the probable losses from Japanese sub- 
marine action shown in Table 2. The second column 
gives the calculated figures; the first, the rounded-off 
estimates. 


Table 1. Torpedo attacks, submarine versus submarine. 



Sunk or 
damaged 

Submarine 

missed 

Attacks on Japanese submarines 

1942 

8 

8 

1943 

4 

11 

1944 

5 

7 

Total 

17 

26 

Attacks on U.S. submarines 

1942 

? 

9 

1943 

? 

11 

1944 

? 

7 

Total 

? 

27 


It turned out that at the time of the analysis, 
reasonable estimates of the effectiveness of Japanese 
antisubmarine planes and ships explained our sub- 
marine casualties only in part, and left approximately 
15 casualties unexplained. The surprising correlation 
between this number 15 and the number 18, esti- 
mated by the above argument to have been sunk by 

Table 2. Estimated losses of U. S. submarines by Jap- 
anese submarine action. 


1942 

6 

(6.0) 

1943 

7 

(7.3) 

1944 

5 

(4.7) 

Total 

18 

(18.0) 


Japanese submarines, made it appear likely that at 
least some of these casualties had been caused by 
Japanese submarines. It was probable that not all the 
15 casualties were due to Japanese submarines, for it 
was likely that the enemy submarines were not as 
effective as ours, which was the assumption made in 
obtaining Table 2. In addition, there must have been 
a certain number of casualties caused by enemy 
mines and by ordinary operational accidents, Avhich 
would account for some of the 15. Nevertheless, the 
above analysis indicated that Japanese submarines 
were likely causes of some of the casualties. Such a 
possibility had not been seriously considered before. 


CONFIDENTIAL 


86 


TACTICAL ANALYSIS 


5.1.9 Suggested Measures 

When this analysis was brought to the attention of 
the higher command, the results suggested the addi- 
tion of certain equipment on our submarines to de- 
tect incoming torpedoes, and certain tactical meas- 
ures (which will be discussed in the next section) to 
protect against this unexpected danger. 

It has since been learned that Japanese subma- 
rines actually sank far fewer than 18 of ours, and that 
many of the unexplained 15 casualties were due to 
enemy mines and operational accidents. Neverthe- 
less, the analysis had indicated a source of danger 
which had been previously minimized, and suggested 
new equipment and tactics to safeguard our subma- 
rines against this danger. After these safeguards had 
been put into use, reports from submarine comman- 
ders indicated that the new equipment and tactics 
probably saved 3 or 4 additional submarines from 
being sunk. 

5.2 ANALYTICAL SOLUTIONS 

INVOLVING SEARCH THEORY 

A great deal of tactical analysis involves the prin- 
ciples of the theory of search. The details of this 
theory are presented in another volume of this series, 
but one theorem is so important for our present dis- 
cussion that it will be worth while discussing it here 
also. This theorem might be called the “mean free 
path theorem,” by its analogy with certain concepts 
in statistical mechanics. 

5.2.1 Covered Area 

The theorem concerns the probability of locating, 
or of damaging, or of colliding with some object, 
called the target, which is placed at random some- 
where within an area A. The search object, which is 
to collide with, or damage, or find the target, can be 
a patrol plane, a torpedo, a bomb, a 16-inch shell, 
etc. Each object has an effective range of action 
against the target: effective range of sighting, lethal 
radius, target width for torpedoes, etc. After a cer- 
tain length of time, some portion of the area A will 
have been covered by one or more of the search 
objects, so that, if the target is within this covered 
area, it will have been discovered or damaged at least 
once. For instance, if the search object is a patrol 
plane, the covered area is equal to twice the effective 
lateral range of vision of the plane, times the speed of 


the plane, times the length of time spent in searching 
the area A. If the test object is a 16-inch shell, the 
covered area is equal to the number of shells fired 
inside A, times the lethal area of the 16-inch shell for 
the target considered. If the search object is a tor- 
pedo, the covered area is equal to the effective width 
of the target ship, times the length of track of the 
torpedo; and so on. 

We assumed that this covered area is distributed 
at random inside the area A. There may be some 
overlap, in that the area is covered more than once, 
but we assume that this is done in a random manner. 
The “mean free path theorem” gives the probability 
of success as a function of the ratio between the 
covered area and the total area A. 

5.2.2 Probability of Hit 

To find the value of this probability, we consider 
the situation at some given instant when the covered 
area is equal to a. We can call P(0, a) the value of the 
probability that the target is not yet discovered or 
damaged before this instant. We then increase the 
covered area by an amount da. If this new covered 
area is placed at random inside the area A, then the 
chance that the target will be found or damaged in 
this new area is equal to the ratio between da and 
total area A, multiplied by the probability that the 
target has not yet been found or damaged before 
this. In other words, 

dP(0, a) = — (da/A)P(0, a ) . 

The solution of this differential equation, which satis- 
fies the initial condition that the probability of no 
hits when a is zero is equal to unity, is the following: 

P(0, a) — e -0 ; 0 = (^j — Coverage factor. 

A = Total area; a = Covered area. (1) 

Probability of hit = P(>0) = 1 — e”*. 

Referring back to equation (28) of Chapter 2, we 
see that the probability of no hits, P(0, a) is just the 
Poisson distribution probability of obtaining zero 
points when the expected number is 0. A little study 
of the relation between the present case and the case 
discussed for the Poisson distribution shows the com- 
plete analogy, and indicates why the coverage factor 
0 is equal to the expected value of the number of 


CONFIDENTIAL 


ANALYTICAL SOLUTIONS INVOLVING SEARCH THEORY 


87 


hits. To carry the analogy farther, we can say that 
the probability that the target will have been dis- 
covered, or damaged, or hit m times when the 
covered area is a is P(m, 0), where 

PK * ) = (S) e " ;0 = (i) ; 

( 2 ) 

CO 

p (>°) = 1 - p (°^) = !-«“*• 

m = 1 

5.2.3 Merchant Vessel Sinkings 

A few simple examples will illustrate the usefulness 
of this theorem. For instance, suppose a merchant 
vessel can make 23 ^ trips, on the average, across the 
ocean before it is sighted by an enemy submarine, 
and suppose that on the average 1 out of every 4 
ships sighted by the submarine is torpedoed. Then 
the “mean free path” of a ship before it gets hit 
would be 10 trips across the ocean; and the expected 
number of hits in n trips would be (n/ 10) . Here, in- 
stead of an area, a line is covered (namely the dis- 
tance covered by the ship) and the coverage factor 0 
is equal to the ratio between the number of trips and 
the “mean free path” (in this case, 10 trips). The 
probability of coming through n trips unscathed is 
P(0, 0), obtained from equation (2), with 0 = (n/ 10). 

5.2.4 Area Bombardment 

Another example can be taken from the study of 
area bombardment. A mortar emplacement, for in- 
stance, is somewhere within an area A. The lethal 
area of the average 5-inch shell for damaging a 
mortar is a. Then if n 5-inch shells are fired at ran- 
dom into the area A, the chance that the mortar will 
be undamaged is e - * , where the coverage factor 0 
equals ( na/A ). The probability of the mortar getting 
hit m times is given in equation (2) . 

Now suppose that the probability that a single hit 
on the mortar will damage it beyond repair turns 
out to be p o, so that the probability of being able to 
repair the mortar after m hits is (1 — p) m . Combina- 
tion of probabilities shows that the overall probability 
of completely disabling the mortar by n shells fired 
into the area A is 

Probability damage beyond repair 

= 1 — probability repairability. 


Probability repairability 

m =0 m = 0 

( 3 ) 

= e~* = e~ p *; <t> = . 

Therefore, the probability of escaping complete de- 
struction can be expressed in terms of a new coverage 
factor 0' = ( pna/A ). This shows that in many cases 
the coverage factor for complete destruction can be 
obtained from the coverage factor for a hit, by multi- 
plying by the probability of complete destruction 
when hit. This simple property of 0 is typical of the 
Poisson distribution. 

Another example of the “mean free path theorem” 
can be taken from the study of the effectiveness of 
mine fields. An influence mine has a range of action 
R for a given ship. If there are n mines in a given 
area A, then the probability that the ship will hit a 
mine is given by 

Probability of hitting at least one mine 



where L is the length of the ship’s track through the 
mine field. The mean free path of the ship in the 
field is thus ( A /2nR ) . 

5.2.5 Antiaircraft Splash Power 

Another example of considerable use in the study 
of antiaircraft defense involves the definition of the 
splash rate for a given battery of antiaircraft guns. 
From the accuracy and rate of fire of these guns, it is 
possible to compute the probability that a given 
plane will be shot down between range r and range 
r + dr. This can be written as s(r)dr, where s(r) is 
called the “splash rate.” Ordinarily this rate is small 
for large values of r and increases as r diminishes. In 
this case the coverage factor 0 is called the splash 
power, which is obtained by integration : 

Splash power, 0(r) = J* s(r)dr; 

s(r) = splash rate. (4) 


CONFIDENTIAL 


88 


TACTICAL ANALYSIS 


The probability that the plane is splashed before it 
reaches range r is 1 — e ~ <t> . By drawing contours of 
constant splash power about the antiaircraft battery, 
one can determine the effectiveness of this battery in 
various directions, and, if necessary, can work out its 
weak points. Since the splash power is additive, the 
powers of different batteries can be added together 
to give an overall contour plot. Contour plots of this 
sort have been useful in determining the correct 
tactics for our own planes against enemy antiaircraft 
fire, as well as evaluating the effectiveness of our own 
antiaircraft batteries. 

5.2.6 Ships Sighted and Sunk by Submarines 

A more complicated example comes from the com- 
parison of the effectiveness of submarines used on 
independent patrol against those used in coordinated 
attack groups. Suppose N submarines are assigned 
to patrol a given shipping lane. Suppose the shipping 
lane has a width W and that each submarine has an 
effective range of vision r. Then by an analysis simi- 
lar to that carried through above, we see that the 
average number S of ships sighted (by one or more 
submarines) per month is 

S = F(1 - e~ 2Nr/w ) . 

Initial sightings per month per submarine 

= Z(l_e-W); (5) 

N 

where F is the total traffic in ships per month. We see 
that there is a definite saturation effect as we increase 
the number of submarines, due to the fact that the 
additional submarines often sight ships which have 
already been sighted. 

When the submarine is on independent patrol, it 
will carry through its attacks separately and will not 
call in the other submarines to help. If P is the prob- 
ability of sinking a merchant vessel once it has been 
sighted, then, by arguments similar to those em- 
ployed in obtaining equation (3), we can compute 
the average number of ships sunk per month by the 
N submarines on independent patrol. 

Hi = F(1 - e ~ 2NPr/w ) 

= No. ships sunk by N submarines on (6) 

independent operation. 


5.2.7 Independent Patrol 

This property of the Poisson distribution, which 
enables us to multiply the coverage factor for sight- 
ing by the probability of sinking once the sighting is 
made, to obtain the coverage factor for sinking, has 
an interesting effect on the saturation of forces. As an 
example of this, suppose we consider the case where 
the shipping lane is twice as wide as the range of 
vision of a submarine. Then, on the average, a single 
submarine would see a fraction 0.63 of all of the ship- 
ping traveling along the lane. Two submarines would, 
on the average, sight 0.86 of all the ships. Conse- 
quently, the addition of the second submarine on 
independent patrol would add to the total number of 
new sightings by only about one-third of the number 
of sightings the first submarine had obtained, an 
example of the saturation effect. This does not mean 
that the second submarine does not see as many 
ships as the first one; it only means that most of the 
ships sighted by the second submarine have already 
been sighted by the first, and that only one-third of 
the second submarine’s sightings are new ones. 

If now, on the average, only one-quarter of the 
ships sighted by the submarine are sunk, we can use 
equation (6) to determine the number of ships sunk 
by a number N of submarines on independent patrol. 
The results are given in the following table : 

On the average: 

( 2r/W = 1; Prob. sinking if sighted = P = 0.25) 

1 submarine sinks 0.22 of shipping flow; 

2 submarines sink 0.39 of shipping flow, the second 
sub giving a gain of 0.77 of the first sub’s catch; 

3 submarines sink 0.53 of shipping flow, the third sub 
giving an added gain of 0.64 of the first sub’s catch; 

4 submarines sink 0.63 of shipping flow, the fourth sub 
giving an added gain of 0.45 of the first sub’s catch; 
and so on. 

Here we see that although the second submarine 
does not make many new sightings, it does account 
for nearly as many additional sinkings as does the 
first submarine. This is due to the fact that the first 
submarine does not sink three-quarters of the ships 
it sights. Therefore, although the second submarine 
usually sights the same ships sighted by the first one, 
it has an additional chance to sink them, which is 
nearly as good as the first submarine. However, as we 
keep on adding submarines the saturation effect 
comes in again, though not as quickly. The fourth 


CONFIDENTIAL 


ANALYTICAL SOLUTIONS INVOLVING SEARCH THEORY 


89 


submarine accounts for less than half the additional 
number one might expect, due to the saturation 
effect. 

5.2.8 Group Operation 

Now suppose these N submarines act together as a 
group, instead of attacking ships independently. In 
this case they will patrol station independently, but 
whenever any submarine sights a ship it will signal 
all the others, who will rendezvous on the submarine 
making the initial sighting and will also attempt to 
sink the ship. We will assume first that all of the sub- 
marines in the group of N manage to home on the 
first one and get their chance at sinking the ship. In 
this case, the probability that the ship is sunk is 
1 — (1 — P) N instead of the value P which it had if 
only one submarine carried out the attack. By the 
same arguments as before, we see that the number of 
ships sunk by a group of N submarines is 

H g = F [l — 

= No. ships sunk by N submarines in (7) 
group operation. 


The relative advantage of group action over inde- 
pendent action is given by the ratio : 



123456789 10 II 


NUMBER OF SUBMARINES, N 

Figure 1 . Relative advantage of group action over in- 
dependent action for submarines against merchant vessels. 

Values of this ratio are plotted in Figure 1. We see 
that, when the shipping lane is narrow, saturation 
soon sets in, and there is a certain optimum size for 
the group. When the shipping lane is very wide, how- 


ever, the advantage continually increases as we add 
more and more submarines to the group, to the degree 
of approximation considered here. 

Actually, of course, other inefficiencies, besides 
saturation, enter as the group gets quite large. Not all 
of the submarines are able to home on the one which 
has made the sighting. The Germans seldom man- 
aged to home more than one-half their pack for 
simultaneous attack, and U. S. submarines in the 
Pacific seldom homed more than one additional sub- 
marine. In addition, when the pack size is large, inter- 
ference would discourage the less aggressive. Conse- 
quently, due to all of these effects, the gain would be 
less than that shown in Figure 1, although it would 
be greater than unity. 

If the shipping travels in convoys, the advantage 
to group action is again increased, for there are a 
number of advantages in combined attack on a 
convoy, some of them mentioned in Section 3.2. 

Although the German “wolf packs” sometimes 
reached a dozen or more, analysis of the sort outlined 
above, using data on Japanese shipping, indicated 
that groups of U. S. submarines of about three per 
group would give optimum results in the Pacific. Fol- 
lowing this analysis, group tactics were tried. After 
the operational tactics had been perfected by prac- 
tice, it turned out that the yield per submarine in a 
group of three was about 50 per cent greater than the 
yield per independent submarine. Thus the analysis 
was borne out in practice. 

5.2.9 Disposition of CAP Protection About 
Task Forces 

Many analyses of tactical problems involve the 
geometrical combination of velocities and tracks 
which also enter into the theory of search. One inter- 
esting example of this comes from the study of the 
proper distribution of combat air patrol [ CAP ] units 
about a task force, to protect the force from enemy 
bomber planes. During World War II, the task force 
itself had the search radar which made the first de- 
tection of the enemy planes. This detection was not 
always made at the same range; there was a certain 
probability distribution F of detection, which de- 
pended on the type of search radar used. To be spe- 
cific, the probability that the enemy unit is detected 
between a range R and a range R + dR is equal to 
F(R)dR. The integral of F over all values of R must 
be unity or less, for this integral equals the fraction 
of enemy planes detected at some range. In actual 


CONFIDENTIAL 


90 


TACTICAL ANALYSIS 


practice, during World War II, this integral was 
nearly equal to unity, for nearly all enemy planes 
were detected before they reached the task force. An 
average curve for this probability of detection F is 
shown in Figure 2. It was obtained from operational 



RANGE IN MILES 



0 50 100 

MEAN PATROL DISTANCE OF CAP UNIT FROM TASK FORCE, p 


Figure 2. Lower curves give average effectiveness, in 
keeping off enemy bombers, of n combat air patrol units, 
each patrolling along a circle of radius about the task 
force. When enemy planes are detected, only the nearest 
unit is vectored to intercept. Curves at top give F, the 
probability of detection of the enemy planes between R 
and R + dR, and E the average per cent loss of enemy 
planes when the interception is made at range r (obtained 
from operational data). 

data taken, during the last year of World War II, 
from actions in the Pacific. 

Detection is not enough, of course; the combat air 
patrol must be vectored to intercept the enemy 
planes and shoot them down. Since the speed of the 
combat air patrol is approximately equal to that of 
the enemy bombers, both enemy and friendly planes 
will have traveled an equal distance between the 
time that the enemy planes are detected and the time 
the patrol planes intercept the enemy. This situation 
is shown in Figure 3, where we have assumed that the 
enemy planes are coming in on a straight course di- 


rected at the task force, and that the nearest combat 
air patrol unit is vectored correctly. Consequently, 
the farther away the CAP unit is from the enemy at 
the instant of detection, the nearer to the task force 
will be the interception. We should place the units so 
they can make interception as soon as possible. 

It is rather obvious that it is desirable to intercept 
the enemy bombers as far away from the task force 
as possible. This gives the CAP unit a longer time to 



Figure 3. Defense of task force from enemy planes by 
combat air patrol units. At instant of detection, nearest 
unit is vectored to intercept. Angle </> is random, since 
bearing of enemy planes is random. 

“work over” the enemy unit and to scatter it or shoot 
it down. Even though the planes are not all shot 
down, a scattered enemy unit seems to find it more 
difficult to get into the task force, perhaps because 
the leader is a better navigator than the other planes 
in the enemy unit and, if the planes scatter, they lose 
their leader. It is of advantage, therefore, to place the 
CAP units in such a way that the interception will 
take place, on the average, as far from the task force 
as possible. The operational data on the fraction of 
enemy planes lost after interception, as a function of 
range of interception, shown in Figure 2, emphasizes 
this point. 

5.2.10 Analysis of Tactical Situation 

The tactical situation, which is to be analyzed, can 
now be stated. The CAP units are made large enough 
to be able to handle the enemy unit without undue 
loss. Suppose that only a number n of such units can 
be kept aloft at the same time. If we do not know the 
direction of the enemy attack, the CAP protection 
should be symmetrically placed, so we assume that 
the units are uniformly spaced around a patrol circle 
of radius p about the task force. At the instant of 
detection the nearest unit is vectored to interception 
as shown in Figure 3. We assume that the speed of 
the CAP unit is the same as that of the enemy planes. 


CONFIDENTIAL 


ANALYTICAL SOLUTIONS INVOLVING SEARCH THEORY 


91 


The range of interception r depends on the angle 0 as 
well as on p and R, according to the equation : 


* R> 


2 (R — p cos 0) 


R > 


p • 


( 9 ) 


Even this simplification does not allow the second 
integration, shown in equation (11), to be accom- 
plished analytically, except for n = 1 . For this special 
case and for a simple assumption concerning the dis- 
tribution function F we obtain, 


We also assume that the fraction of enemy planes lost, 
when an interception takes place at a range r from 
the task force, is E(f). An approximate curve for E, 
obtained from operational data, is shown in Figure 2. 

At the instant of detection the CAP unit, which is 
vectored to the interception, happens to be at the 
position corresponding to the angle 0, where 0 is less 
in magnitude than (jr/ri) (or else this unit would not 
be the nearest one to the enemy planes) . If the enemy 
planes are equally likely to come in from any direc- 
tion, any value of 0 is possible between the limits 
(tt/ ri) and —(r /ri). Consequently, the average value 
of E, the fraction of enemy planes not getting through 
after the interception is given by 

r/n 

E(r)d4> R > p 

n 

( 10 ) 

P • 

However, ranges of first detection are not always the 
same, but vary according to the distribution function 
F(R). The average value of the fraction of enemy 
planes lost by CAP interception for all values of de- 
tection range is given by 

w n (p) =J° F{R)E n (R, p)dR , (11) 




The probability function F starts at zero for R = 0, 
rises to a maximum at R = Ro, and then approaches 
zero again for very large values of R. 

The average effectiveness of a single CAP unit is 
given by Wi, where the function K 2 is the Bessel 
function of the second kind, of imaginary argument, 
and of second order, as defined by Watson. 13 This 
function is plotted as a dashed curve in the lower set 
of curves in Figure 2, for R Q = 25 miles. We note that 
its maximum value is at p = 0, indicating that, if 
there is only one CAP unit which can be kept aloft 
at a time, it is most effective to keep this unit directly 
above the task force, as long as one does not know 
the direction from which the enemy planes are likely 
to come, for if the single unit were patrolling at a 
distance from the task force it might get caught “off 
base” by an enemy unit coming in to the task force 
from the opposite side. This result turns out to be 
true for other reasonable assumptions as to E and F: 
if only a single CAP unit is aloft at a given time, its 
most effective position is directly over the task force 
(unless the direction of enemy attack is known). 


which might be called the effectiveness of the CAP 
patrol disposition . 


5.2.11 A Simple Example 


These calculations cannot be carried through ana- 
lytically unless the functions E and F are extremely 
simple. If we rate the effectiveness of an interception 
as a linearly increasing function of the range of inter- 
ception r, then the first step in the calculation can be 
carried through analytically. 

Assume E(r) = ( r/R m ); then 



p—R cos (r/n) 
_R — p cos (j /n) 



5.2.12 Several CAP Units 

The integration for the more general case, with 
several CAP units aloft, must be carried through nu- 
merically. Consequently we might as well use the 
curves for E and F obtained from the operational 
data, shown in Figure 2. The results of this calcula- 
tion are shown in the lower set of solid curves in 
Figure 2. The maximum values of these curves indi- 
cate the optimum radius of the patrol circle for the 
n units of combat air patrol. We notice that, for the 
curves of E and F used : if only one CAP unit can be 
kept aloft at a time, it should patrol over the task 
force; if two units can be kept aloft, they should pa- 
trol on opposite sides of the task force and about 20 
miles away from the task force; if three units are 
aloft, they should patrol on a circle of radius 25 miles, 


CONFIDENTIAL 


92 


TACTICAL ANALYSIS 


spaced 120 degrees apart along this circle; etc. Even 
for an extremely large number of CAP units, if these 
are distributed on a single circle, the diameter of the 
circle should only be 35 miles in radius, equal to 
about one-half the mean range of detection. 

The discussion presented here is only the beginning 
of the complete tactical study. One must investigate 
the possibilities of vectoring out a second unit to 
“back up” the first unit, for it sometimes happens 
that the first unit does not make an interception 
effectively. This can be taken into account to some 
extent by successive use of the function E, but, 
strictly speaking, the mean free path theorem should 
be used to obtain a more detailed answer. In many 
cases, also, it is more likely that the enemy units will 
approach from one side rather than another. In this 
case, the integration over the angle <f> must include 
the probability of the enemy units coming in from a 
given direction. The results would indicate how the 
disposition of CAP units would have to be modified. 
The present calculations also do not include the effect 
of the altitude of the enemy units on the interception 
problem. Many of these aspects have been dealt with 
in various ORG studies; space cannot be given to 
them here. 

5.2.13 Tactics to Evade Torpedoes 

The last example given in this section will continue 
the analysis of the submarine versus submarine prob- 
lem discussed in Section 5.1. There it was shown that 
there was a possibility that our own submarines in 
the Pacific were being torpedoed by Japanese sub- 
marines, and that there was a good chance that 
several of our casualties were due to this cause. Pre- 
sumably the danger was greatest when our subma- 
rine was traveling on the surface and the enemy 
submarine was submerged. It was important, there- 
fore, to consider possible measures to minimize this 
danger. One possibility was to install a simple under- 
water listening device beneath the hull of the sub- 
marine, to indicate the presence of a torpedo headed 
toward the submarine. Torpedoes driven by com- 
pressed air can be spotted by a lookout, since they 
leave a characteristic wake; electric torpedoes, on 
the other hand, cannot be spotted by their wake. All 
types of torpedoes, however, have to run at a speed 
considerably greater than that of the target, and 
therefore their propellers generate a great deal of 
underwater sound. This sound, a characteristic high 
whine, can be detected by very simple underwater 


microphones, and the general direction from which 
the sound comes can be determined by fairly simple 
means. 

Microphone equipment to perform this function 
had already been developed by NDRC; it remained 
to determine the value of installing it. In other words, 
even if the torpedo could be heard and warning given, 
could it be evaded? The chief possibility, of course, 
lay in radical maneuvers.. A submarine (or a ship) 
presents a much smaller target to the torpedo end on 
than it does broadside. Consequently, as soon as a 
torpedo is heard, and its direction is determined, it is 
advisable for the submarine to turn toward or away 
from the torpedo, depending on which is the easier 
maneuver. 

^•2.14 Geometrical Details 


The situation is shown in Figure 4. Here the sub- 
marine is shown traveling with speed u along the 



/ 


Figure 4. Quantities connected with analysis of torpedo 
attack on submarine or ship. 

dash-dot line. It discovers a torpedo at range R and 
at angle on the bow 6 headed toward it. For correct 
firing, the torpedo is not aimed at where the subma- 
rine is, but at where the submarine will be when the 
torpedo gets there. The relation between the track 
angle <f>, the angle on the bow 6, the speed of torpedo 
and submarine, and the range R can be worked out 
from the geometry of triangles. The aim, of course, 
is never perfect, and operational data indicate that 
the standard deviation for torpedoes fired from U. S. 
submarines is about 6 degrees of angle. 

In most cases, more than one torpedo is fired. For 


CONFIDENTIAL 


ANALYTICAL SOLUTIONS INVOLVING SEARCH THEORY 


93 


instance, if three torpedoes are fired in a salvo, the 
center torpedo is usually aimed at the center of the 
target. If the other two are aimed to hit the bow and 
stern of the target,* the salvo of three is said to have a 
100 per cent spread. Due to the probable error in aim, 
it turns out to be somewhat better to increase the 
spread to 150 per cent, so that, if the aim were per- 
fect, the center torpedo would hit amidships, and the 
other two would miss ahead and astern. Analysis of 
the type to be given in Chapter 6 shows that a salvo 
of three with 150 per cent spread gives a somewhat 
greater probability of hit than does a salvo with 100 
per cent spread. 

A glance at Figure 4 shows that if the track angle <f> 
is less than 90° the submarine should turn as sharply 
as possible toward the torpedoes in order to present 
as small a target as possible; if the track angle is 
greater than 90° the turn should be away from the 
torpedoes. Assuming a three torpedo salvo, with 150 
per cent spread and 7° standard deviation in aim, and 
knowing the maximum rate of turn of the submarine 
and the speed of the submarine and torpedo, it is 
then possible to compute the probability of hit of the 
salvo, as a function of the angle-on-the-bow 6 and 
the range R at which the submarine starts its turn. 
If the range is large enough, the submarine can turn 
completely toward or away from the torpedoes (this 
is called “combing the tracks”) and may even move 
completely outside of the track of the salvo. If the 
torpedoes are not discovered until at short range, 
however, very little improvement can be obtained by 
turning. 

One can therefore compute the probability of hit- 
ting the submarine if it starts to turn when it hears a 
torpedo at some range and angle-on-the-bow. This 
can be plotted on a diagram showing contours of 
equal probability of sinking, and these can be com- 
pared with contours for probability of sinking if the 
submarine takes no evasive action, but continues on 
a straight course. A typical set of contours is shown 
in Figure 5. 

The solid contours show the probabilities of a hit 
when the submarine takes correct evasive action. 
The dotted contours give the corresponding chances 
when a submarine continues on a steady course. One 
sees that the dotted contour for 30 per cent chance of 
hit covers a much greater area than a solid contour 
for the same chance. In other words, at these longer 
ranges the evasive action of the submarine has a 
greater effect. The contours for 60 per cent chance of 
hit do not show the corresponding improvement, 


since, by the time the torpedo is so close to the sub- 
marine, maneuver has little chance of helping the 
situation. One sees that, if one can hear the torpedo 
as far away as 2,000 yards, a very large reduction in 
the chance of being hit can be produced by the cor- 
rect evasive maneuvers. 

Since these contours represent, in effect, vulnera- 
bility diagrams for torpedo attack, they suggest the 
directions in which lookout activity should be em- 



500 1000 1500 

SUBMARINE 


RANGE OF DETECTED 
TORPEDO IN YARDS 

U S FLEET TYPE SUBMARINE AT 18 KNOTS 
3 TORPEDO SALVO (150% SPREAD) AT 45 KNOTS 

Figure 5. Chance of surviving torpedo salvo by sharp 
turns as soon as torpedo is detected, as function of torpedo 
range and bearing when detected, compared to chance of 
survival when no evasive action is taken. 

phasized. The greatest danger exists at a relative 
bearing corresponding to a 90 degree track angle, 
and the sector from about 30 degrees to 105 degrees 
on the bow should receive by far the most attention. 
The narrow separation of the contours corresponding 
to evasive action emphasizes the extreme importance 
of the range of torpedo detection. In many instances 
a reduction of 500 yards in detection range may cut 
in half the target probability of escaping. 

Another factor vital to the efficacy of evasive 
turning is the promptness with which it is initiated. 
For 45-knot torpedoes, every 10 seconds delay in exe- 
cution of the turn corresponds approximately to a 
reduction of 250 yards in the distance from the tor- 
pedo to the target. Thus it is apparent that a 20 
seconds delay in beginning the evasive turning will 
probably halve the chances of successful evasion. 

These same calculations, with different speeds and 
different dimensions for the target vessel, may be 
used to indicate to the submarine where it is best to 


CONFIDENTIAL 


94 


TACTICAL ANALYSIS 


launch its torpedoes in order to minimize the effect 
of evasive turning of the target ship. One sees that 
it is best to launch torpedoes, if possible, with a track 
angle of approximately 90 degrees. One sees also the 
importance of coming close to the target before firing 
the salvo, since evasive action is much less effective 
when begun with the torpedo less than 2,000 yards 
away. 

This study showed the value of good torpedo- 
detection microphones, with ranges of at least 2,000 
yards, and supported the case for their being installed 
on fleet submarines. Publication of the study to the 
fleet indicating the danger from Japanese submarines 
and of the usefulness of evasive turns, produced an 
alertness which saved at least four U. S. submarines 
from being torpedoed, according to the records. 

5.3 MEASURE AND COUNTER- 

MEASURE 

Some of the most urgent tasks and the most excit- 
ing opportunities for operations research lie in the 
field of the devising of countermeasures to new enemy 
tactics or weapons. Nearly every aspect of World 
War II showed an interplay of measure and counter- 
measure : the side which could get a new measure into 
operational use before the enemy realized what it 
was, or which could get a countermeasure into use 
before the enemy had perfected his methods of using 
some measure, was the side which gained tremen- 
dously in this interplay. Operations research workers 
helped considerably in speeding up these reactions, 
by following technical developments closely and by 
relating them to the most recent operational data. 
In a narrow sense, of course, this is not operations re- 
search, but operations research workers proved most 
useful, since they were familiar with the tactical 
situation. 

Most of the operational decisions and planning on 
countermeasures require a great deal of technical 
background. Information from espionage and other 
intelligence sources often comes through in fragmen- 
tary form, and unless the persons analyzing the in- 
formation know the technical possibilities, only a 
fraction of the important information may ever be 
discerned. Knowledge of new enemy measures must 
then be carried to those laboratories which are cap- 
able of turning out countermeasures quickly and 
accurately. Since it often happens that the weak 
points of an enemy measure are things which could 
easily be remedied by the enemy if he thought it 


necessary, it is usually quite important to keep our 
knowledge of such information at a high security 
level. The problem of introducing enough technical 
men to the intelligence information in order to solve a 
problem rapidly, while maintaining proper security, 
is one of the reasons these problems are difficult ones. 

5.3.1 Countermeasures to Acoustic Torpedoes 

The first information on the German T-5 Acoustic 
Torpedo came from espionage. The first information 
which was of technical value came from fragmentary 
descriptions of the torpedo by prisoners. By piecing 
together these descriptions, a fairly sensible picture 
of the design was obtained, and, by using the two 
available guesses as to size, the dimensions of impor- 
tant units could be estimated roughly. The problem 
was serious enough to warrant requesting a labora- 
tory to build an acoustic control head according to 
the estimated specifications. In the meantime, calcu- 
lations involving the properties of diffraction and of 
acoustic resonance were utilized in order to obtain a 
preliminary estimate of the acoustical behavior of 
this torpedo. Combining measurements on the sam- 
ple built by the laboratory, theoretical calculations, 
and further detailed intelligence information made it 
possible to obtain a rough estimate of its character- 
istics. 

The important characteristics were the speed of 
the torpedo; its turning radius; the extent of the 
region around the torpedo within which the hydro- 
phones were sensitive to sound; the frequency for 
optimum response of hydrophones; and the sensitiv- 
ity of response in the steering mechanism to changes 
of direction of the sound source. The torpedo was 
to be fired from a considerable distance and to travel 
as an ordinary torpedo for the majority of its run. 
The hydrophones were then turned on, the speed of 
the torpedo was reduced to reduce self-noise, and the 
torpedo steered toward whatever noise source was in 
front of it. 

Since the sensitive element was a pair of hydro- 
phones, which could only tell whether the torpedo 
was steering to the right or left of the source, the 
torpedo probably would follow a pursuit course, not 
a collision course. On a pursuit course the torpedo 
constantly points toward the source of sound, which 
the launcher hopes will be the ship’s propellers. If the 
torpedo is initially forward of the ship, the torpedo 
path will exhibit a greater and greater curvature 
until the torpedo turns around the stern of the ship 


CONFIDENTIAL 


MEASURE AND COUNTERMEASURE 


95 


and begins a stern chase (unless the torpedo track 
is so nearly a collision track that it hits the hull of 
the ship as it goes b.y) . If the track angle is large, the 
greatest curvature of this pursuit path may be less 
than the maximum curvature possible for the tor- 
pedo. In many cases, however, the torpedo will not 
be able to turn sharply enough to follow the pursuit 
course. At this point, whether the torpedo eventually 
swings back on the ship’s track to complete its stern 
chase, or whether it loses the target completely, de- 
pends on how concentrated, around the bow direc- 
tion, is the directional listening pattern of the tor- 
pedo’s hydrophones. 

The obvious countermeasure to such a torpedo is 
to tow after the ship an underwater noisemaker, 
which is enough louder in the proper frequency range 
than the propellers, so that the torpedo will steer 
for the noisemaker rather than for the propellers. 
Noisemakers could be tossed overboard to drift 
astern, but this would require too large an expendi- 
ture of material, so it was preferable to tow the noise- 
maker, if this would provide sufficient protection. 
Pursuit curves, for various intensities of the noise- 
maker and for various distances astern of the ship, 
had to be computed, using different reasonable 
assumptions concerning the spread of the directional 
listening pattern of the torpedo and its range of 
acuity. On the basis of these calculations, it was de- 
cided that a single noisemaker, towed some distance 
astern, would provide reasonable protection against 
a torpedo with the acoustic and control properties 
which seemed most probable. The specifications also 
required a certain minimum intensity of the noise- 
maker in the important frequency range, which by 
that time had been determined to be within 10 and 
15 thousand cycles per second. 

The experimental tactical unit of the Antisubma- 
rine Development Detachment, Atlantic Fleet, was 
then utilized to make full-scale measurements on 
various types of noisemakers. The parallel-pipe vi- 
brators called FXR turned out to be as loud as most 
and to be somewhat easier to handle than most. By 
this time a working model, estimated to correspond 
to the German torpedo, was built and could be used 
to verify the calculations. The results were satisfac- 
tory and the countermeasure gear, the FXR, was 
supplied to the antisubmarine craft for their protec- 
tion, together with the necessary doctrine for its use. 

Most of these calculations and tests had already 
been finished by the time the Germans came out with 
the T-5 torpedo in operation. A few destroyers were 


sunk by the torpedo before the countermeasure gear 
could be supplied; but no destroyers were hit by T-5 
torpedoes when they were towing the noisemaker ac- 
cording to doctrine, although many acoustic torpedoes 
were fired at such destroyers (and several noisemakers 
were blown up by direct torpedo hits !) . The German 
U-boat Command was greatly disappointed at the 
rapidity with which this countermeasure was gotten 
into use and the consequent failure of their new tor- 
pedo. 

5.3.2 Radar Countermeasures 

The radar field was the greatest arena of counter- 
measures in World War II, and the struggle reached 
its greatest complexity in the aspects connected with 
strategic bombing. The activities in this field are suf- 
ficiently complex to require several volumes to ex- 
plain them, and space forbids their discussion here. 
The radar countermeasure struggle in the anti-U- 
boat campaign was a comparatively simple one, but 
it demonstrates most of the elements of the problem, 
and will be discussed here for its simplicity. A pre- 
liminary discussion of this phase has already been 
given at the end of Section 3.1. 

Radar has been only one of the many weapons 
applied to counter the enemy use of U-boats, but it 
played an important role at certain critical times and 
caused grave concern to the U-boat Command. The 
moves and countermoves of the radar war offer an 
interesting example of the importance of quick and 
accurate evaluation of enemy measures, and of the 
operational effectiveness of enemy countermeasures. 
Only rarely is a countermeasure widely enough ap- 
plied or sufficiently effective to justify the extreme 
tactic of abandoning the weapon; usually the prompt 
application of counter-countermeasures will restore 
effectiveness. This is particularly true of the radar vs 
search-receiver competition, which was a continuing 
problem throughout the U-boat Avar. 

From the start of the war the Germans were fully 
aware of the possibilities of meter ASV radar and 
had developed their own airborne search equipment. 
When, in the summer of 1942, they concluded cor- 
rectly that the Allies were using radar for U-boat 
search, they initiated a “crash” program for the 
development of search receivers [GSR] to detect 
the radiations. The first equipment to be installed on 
U-boats was the R600 or “Metox,” with a low- 
wavelength limit of 130 cm. It was of the heterodyne 
type, thought to be the only type capable of sufficient 


CONFIDENTIAL 


96 


TACTICAL ANALYSIS 


sensitivity, and it radiated energy: in fact, if it had 
been designed as a transmitter it could hardly have 
radiated more power. Its operational success against 
the British Mark II radar was undeniable, and it was 
accepted as a satisfactory warning receiver by U-boat 
captains. 

Meanwhile, Allied development of S-band radar 
was proceeding, based upon the magnetron trans- 
mitter tube, and was put into operational service in 
early 1943 as the U. S. ASG and the British Mark III 
types. From the start this met with operational suc- 
cess and U-boat sinkings increased. The Germans be- 
came convinced that Allied aircraft were using some 
new detection device, and started a frantic activity 
to identify and counter it. For a time they occupied 
themselves with the idea that it was an infrared de- 
tector, and experimented with their own infrared 
detectors and with special paints intended to give no 
infrared reflections. They also considered the possi- 
bility of a frequency-scanning radar and developed a 
scanning receiver with a cathode-ray tube presenta- 
tion. This was of definite advantage to the operator, 
but it still covered the same meter- wave band. 

The sinkings of U-boats continued. In desperation, 
they jumped to the conclusion that their GSR radia- 
tions were being homed on. The Metox receiver was 
outlawed, and the “Wanz” Gl introduced. This was 
of an improved design and radiated much less power. 
However, the almost pathological fear of radiations 
which had been bred in the minds of U-boat captains 
prevented them from trusting it. Continued sinkings 
and skepticism of the technical advantages kept it 
from being used. Next, the German scientists turned 
to the much less sensitive crystal detector receiver, 
which was entirely free from radiation, and produced 
the “Borkum.” This was a broad-band intercept re- 
ceiver which covered the 75-300 cm band. 

Finally, in September 1943, the U-boat Command 
recognized that 10 cm band radar was being used for 
U-boat search. One of these sets had been captured 
intact at Rotterdam by the German Air Force in 
March 1943, and German scientists had soon deter- 
mined its characteristics but the news reached the 
German Navy in September. How this six months’ 
delay occurred is one of the mysteries of the war and 
a significant factor in the U-boat war (it can perhaps 
be explained only by a criminal lack of liaison be- 
tween the German air and naval technical staffs). 
A further delay of six months intervened before the 
first effective S-band receivers became operational, 
in April, 1944. During this interval the frantic ex- 


perimentings of the German Technical Service be- 
came evident in such incidents as the patrol of the 
U-406 carrying one of their best GSR experts, Dr. 
Greven, and his staff, with a full complement of 
experimental search receivers. The U-406 was sunk, 
and Dr. Greven was captured. Other experimental 
patrols had even shorter careers. 

5 - 3.3 N axos Search Receiver 

Out of this confusion finally came the “Naxos” 
intercept receiver covering the 8-12 cm band. The 
first models were crude, portable units mounted on a 
stick and carried up through the conning tower on 
surfacing. The range was short, due to the crystal 
detector principle, the broad-band coverage and the 
small, nondirectional antenna; estimates of range 
from P/W reports are 8 to 10 miles. The equipment 
was subject to rough handling on crash dives and was 
often out of order. Continued development improved 
the reliability, and it eventually proved its value in 
giving warning of Allied S-band radar, usually at 
ranges about equal to radar contact ranges. This re- 
sulted in an increase in the number of “disappearing 
contacts” on the radars and an even greater number 
of successful evasions which can only be estimated. 

Allied reaction to intelligence reports about Naxos 
as early as December 1943 brought the fear that S- 
band radar was compromised. A serious morale prob- 
lem developed among Allied ASV flyers with this 
news and with the related drop in U-boat contacts. 
Radar was turned off completely in several squad- 
rons. Tactics were improvised to salvage some useful- 
ness for the radars, on the assumption that the GSR 
could outrange the radar. On the approach “search- 
lighting” the target, sector scan, or change of scan 
rate were not allowed, since such changes would 
indicate to the GSR operator that radar contact had 
been made, and the U-boat could take evasive action. 
Attenuators, such as “Vixen,” were initiated to cause 
a slow and steady decrease in transmitted power as 
range closed, so as to confuse the GSR operator. In 
order to use this successfully, the contact must be 
made at a range of 15 miles or greater. Since this was 
greater than average radar contact range under many 
conditions, it could only be used for about half the 
contacts. Production was slow and installations 
slower; Vixen never did have an operational oppor- 
tunity of justifying the effort spent in its develop- 
ment. An interim tactic of a “tilt-beam” approach to 
reduce signal intensity as range closed was proposed. 


CONFIDENTIAL 


MEASURE AND COUNTERMEASURE 


97 


This required unusual skill and cooperation between 
pilot and radar operator to be effective, and its 
value was never adequately proven. Almost in des- 
peration the tactic of turning the spinner aft (for the 
360-degree scanning radars) and approaching by 
dead reckoning was suggested. The chances of a suc- 
cessful navigational approach were small, however, 
as compared to radar homing on the target. 

5.3.4 Allied Reaction 

The chief error made by the Allies at this phase was 
in overestimating the capabilities and efficiency of the 
Naxos GSR. Analyses of sighting data, mentioned in 
Section 3.1, soon showed that the GSR was far from 
being certain protection for the U-boat. Efforts were 
made to revive the confidence in radar and keep it in 
operation. The validity of this view was indicated by 
the continued high rate of U-boat sinkings up through 
August 1944, when the withdrawal from French 
coastal ports caused a large drop in U-boat activity. 

The use of radar of an even higher frequency was 
an obvious next step. Development and allocations 
of X-band equipments even preceded the advent of 
Naxos, and were further stimulated by the problem 
it presented. However, the Germans were not caught 
napping this time. An H2X blind-bombing A/C was 
lost over Berlin in January 1944, and from the dam- 
aged remains the Germans learned of the frequency 
band. It was assumed that this frequency would also 
be applied to ASV radar, and the development of 
X-band intercept receivers was started even before 
use of X-band radar by the Allies in U-boat search 
became operationally effective. A well-designed re- 
ceiver, known as “Tunis,” which consisted of two 
antennas, the “Muecke” > horn for X-band and the 
“Cuba la (Fliege)” dipole and parabolic reflector for 
S-band, was developed and installations started in 
the late Spring of 1944. Installations seem to have 
been completed during the period of inactivity fol- 
lowing the withdrawal to Norwegian and German 
bases. Two amplifiers with a common output to the 
operator’s earphones made it possible to search both 
bands simultaneously. The chief feature was the di- 
rectional antennas, which gave increased sensitivity 
and range; the range probably exceeded radar con- 
tact range for all X and S radars of that time. To 
obtain full coverage the antennas were mounted in 
the D. F. loop on the bridge and rotated manually at 
about two revolutions per minute. The unit still was 
to be dismounted and taken below on submerging, 


and so could be used only in the surfaced condition. 
It seems to have been a reliable and effective warning 
receiver. , 

5.3.5 Intermittent Operation 

Intermittent operation of ASV radar might have 
become a valuable counter to such directional re- 
ceivers. A schedule of two or three radar scans at 
intervals of one to two minutes for a narrow-beam 
radar will point the beam “on target” for only short 
time intervals. The probability of detection is de- 
termined by the chance of coincidence of these time 
intervals with the intervals when the receiver antenna 
is directed toward the radar. Knowledge of the radar 
and GSR beam widths and scan frequencies make it 
possible to compute the probability of detection per 
minute, Pi, for each intermittency schedule. The 
cumulative probability of detection in the time re- 
quired for the radar aircraft to approach from GSR 
range to average radar contact range is given by P t = 
1 — (1 — Pi)*. The probability of undetected ap- 
proach to a radar contact (Q t = 1 — Pt) can be made 
as high as 70 per cent by proper choice of the inter- 
mittency schedule. A small reduction in radar contact 
efficiency or sweep width is to be expected, but is, in 
general, much less than the loss in search receiver- 
detection probability, and the result is net gain. 

The above tactic of intermittent use of radar is of 
most value against highly directional search re- 
ceivers such as Tunis. All-round-looking receiver 
antennas will not be countered to the same extent. 
However, the psychological confusion of the re- 
ceiver operator in interpreting the short and infre- 
quent signals will result in a definite but incalculable 
reduction in efficiency. Furthermore, the shorter 
range and reduced sensitivity of the nondirectional 
antennas will mean that a shorter time interval is 
involved. So there may be advantages of intermit- 
tent radar operation even for such nondirectional 
search receivers. 

One of the most important results of the intensive 
Allied use of search radar was in driving U-boats 
under the surface and so, in blinding and partially 
immobilizing them, reducing their effectiveness. 
Hold-down, or flooding, tactics to achieve this end 
are of recognized value for convoy coverage and in 
congested areas. Radar will no doubt continue in 
use even though a future GSR of greater sensitivity 
and more perfect coverage is produced, in order to 
prevent U-boats from again adopting surfacing 


CONFIDENTIAL 


98 


TACTICAL ANALYSIS 


tactics. Furthermore, no device is ever 100 per cent 
efficient operationally and will occasionally fail. Con- 
tinued use of radar will capitalize on this inefficiency 
and will result in some successful contacts. 

5.3.6 Estimate of Effectiveness of Enemy’s 
Measures 

The preceding discussion of radar countermeasures 
illustrates one of the most important problems for 
operations research in the field of countermeasures: 
namely, the proper estimation of the time to intro- 
duce the countermeasure. It was pointed out above 
that there was a tendency among the Allied anti- 
submarine forces to turn off their microwave radar 
before the German microwave search receiver had 
become effective. Thus the Allied antisubmarine air- 
craft were reduced by a factor of 2 or 3 in effective- 
ness before it was really necessary to make the reduc- 
tion. A detailed comparison between visual and radar 
contacts in the Western Atlantic showed that there 
was little actual reduction in the ratio of visual to 
radar sweep rate until the end of the war. Therefore, 
even if the Germans were using their search receiver, 
it was not doing them much good at that time, and 
there was no reason to hamper our own radar search 
aircraft by introducing countermeasures until effec- 
tiveness had improved. 

This situation is typical of a great number of cases. 
There are indications that the enemy has begun, or 
is likely to begin, the use of a countermeasure which 
may destroy the effectiveness of one of our measures. 
We have in turn developed a counter to this which 
may or may not reduce the effectiveness of the 
enemy’s countermeasure, but which is detrimental to 
our measure unless the enemy is using its counter- 
measure. In a few cases, the effects of the enemy’s 
countermeasure are so apparent that we can nearly 
always tell when he uses it. We can then follow the 
situation, and can introduce our own counter when 
the enemy uses his countermeasure a great enough 
percentage of times to make our counter worth while. 

In a great number of cases, however, we cannot be 
sure in each encounter whether the enemy was using 
his countermeasure or whether he was just lucky in 
that particular case. A certain percentage of the time 
the enemy’s countermeasure is not used and our 
measure is effective, in another per cent of the time 
our measure fails even though the countermeasure is 
not used; in a part of the time the enemy’s counter- 
measure is used but is not effective, and the re- 


mainder of the time the enemy’s countermeasure is 
used and is effective. In such cases, we are not as 
interested in knowing what percentage of the time 
the enemy uses his countermeasure as we are in 
knowing whether our countermeasure would be able 
to help the situation. 

Such a question can only be answered by trial in 
operation. Each month we try a certain number of 
times using the countermeasure, and the remainder 
of the times we do not use it. If the results show that 
the counter to the enemy’s countermeasure gives 
better results, we then use it; if it does not, we try 
again later to see whether the enemy has improved 
his countermeasures. Since all such trials are ran- 
dom affairs, we must be sure that our results have 
meaning statistically. Consequently, it is well to pro- 
vide criteria for determining when our tests have 
meaning. 

5.3.7 Discrete Operational Trials 

There are two cases which must be considered 
separately. The first is where the operation consists 
of a discrete try, such as a firing of a torpedo or 
guided missile. The other case is where the operation 
involves continuous effort, such as the aircraft search- 
ing for a submarine, or the submarine waiting for a 
ship. The first case can be exemplified by the follow- 
ing example: we have been using air-launched, anti- 
ship guided missiles against the enemy with fair suc- 
cess. This success has recently been reduced, which 
leads us to suspect that the enemy is using certain 
jamming methods which disturb the homing mecha- 
nism in the guided missile. We have developed an 
antijamming device which can be inserted in the 
homing mechanism of our missiles. This device is 
complex enough so that in a certain number of cases 
the homing mechanism will break down and fail. 

On the other hand, when it does not fail, it will 
counteract the enemy’s jamming equipment in a 
certain percentage of the cases. We are sure that if 
the enemy is not using jamming equipment, the 
antijamming equipment would be a detriment to in- 
stall. If the enemy is using jamming equipment 
enough of the time, however, it probably would be 
best to install the antijamming mechanisms. We must 
make a series of tries with and without the anti- 
jamming equipment in order to see which is the best 
result, on the average. Since the enemy is probably 
changing his tactics from time to time, we must 
continue to make these tests; at the same time, how- 


CONFIDENTIAL 


MEASURE AND COUNTERMEASURE 


99 


ever, we must arrange our actions so that the ma- 
jority of the time we use that operation which we 
believe will give best results. 


5 . 3.8 Mathematical Details 

To see what should be done we first consider the 
general case where we have made n trials without the 
antijamming equipment (Tactic 1) and N trials using 
the antijamming equipment (Tactic 2). Suppose in 
s of the n trials without antijamming equipment, we 
are successful (i.e., the guided missile sinks a ship), 
and in S cases, out of the N tries with Tactic 2, we 
have success. Then, if ( s/n ) is larger than ( S/N ), our 
information would lead us to think that the enemy’s 
countermeasure was not effective enough to make it 
worth while to install antijamming equipment yet. 
However, the results we have actually obtained 
might be due to fluctuation and might not represent 
the average case. We should like to determine, by our 
series of tests, the values of the probability of success p 
and P of the two tactics. If p is larger than P, then we 
should definitely use Tactic 1; if ( p/P ) is smaller 
than unity, we should use Tactic 2 (antijamming 
device) . 

If we actually knew the values p and P, we could 
compute the probability of obtaining the result we 
did. From Chapter 2, equation (14), we see that this 
probability is 


nJNJ 


s \ (n — s) \ S \ (N — S) \ 


ps(l-p)n-spS(l_p)N-S 

= Mp,P). 


Unless n and N are both small, this expression is a 
rather difficult one to handle. In general, however, we 
will have to make enough trials to be sure of our 
answer, so that n and N will not be small. If these 
quantities are not small, however, we can use the 
approximation methods discussed in obtaining equa- 
tion (24) of Chapter 2. These same methods give the 
approximate result : 

, , nW 3 

d ' V ’ _ 4ir 2 sS(n-s) ( N-S ) 

--V — (p-*Yl. 

2s(n — s) V n) 2S(N-S)\ N/ J 

This probability has a maximum at p = (s/n) and 
P = (S/N), as shown in Figure 6. In terms of this 



figure, we see that our question is as follows: We have 
obtained results s and S; what is the probability that 
p is larger than P? From the figure we see that this 
must equal the integral f(p, P) over all the space to 
the right of the diagonal dashed line. A great deal of 
algebra is needed to show that this probability is 

Prob. p > P 

where F n is the function given in equation (24) of 
Chapter 2. According to Figure 11 of Chapter 2 this 
probability is 50 per cent if the quantity in the 



Figure 6. Calculation of probability that Tactic 1 or 

Tactic 2 is more successful. 

braces is zero; it is approximately 10 per cent if this 
quantity is — 1.4; and it is 90 per cent, approximately, 
if the quantity is +1.4. 

We can say that, if s, S, n, N have such values that 
this probability (that Tactic 1 is better than Tactic 
2) is less than 1 chance in 10, we would naturally 
prefer to use Tactic 2 (antijamming device). Since 
the enemy is likely to change his countertactics, 
however, it is well to keep a certain percentage of 
Tactic 1 going in order to keep a continuous check. 
If the probabilities of Tactic 1 being better than 
Tactic 2 are less than 1 chance in 10, however, we 
should not use Tactic 1 very often; it should be used 
less than one-tenth of the time, as a matter of fact. 
Similarly, if the probability P n is larger than 90 per 
cent, we should not use Tactic 2 any more than 1 in 
10 times, etc. 


CONFIDENTIAL 


100 


TACTICAL ANALYSIS 



0 0.5 1.0 


RATIO SUCCESSES TO TRIALS FOR TACTIC 1, (s/n) 

Figure 7. For s successes out of n discrete trials of Tactic 1, S successes out of N trials of Tactic 2, a point is deter- 
mined on the plot. Rules in text tell how future action should depend on position of point with respect to contours. 


5 - 3 - 5 * * * 9 Rules for Trials 

With this sort of reasoning in mind, we proceed to 
make rules of procedure, which of necessity must be 

more clear-cut than the probabilities ever can be. 

These rules, which nevertheless give a fairly good 
approximation to the discussion of the last para- 
graph, are as follows: 

1. If the quantity in the braces in equation (12) is 
less than —1.4, use Tactic 1 one-tenth as often as 
Tactic 2. 


2. If the quantity in the braces is between —1.4 
and 0, use Tactic 1 one-half as often as Tactic 2. 

3. If the quantity is between 0 and +1.4, use 
Tactic 1 twice as often as Tactic 2. 

4. If this quantity is larger than +1.4, use Tactic 1 
10 times as often as Tactic 2. 

5. If situation (1) or (4) continues for several 
months, and if other intelligence indicates that it is 
likely to continue, discontinue Tactic 1 or Tactic 2 
entirely. 

These requirements can be presented graphically, 


CONFIDENTIAL 



MEASURE AND COUNTERMEASURE 


101 


in Figure 7. From the results of the week’s (or 
month’s) trials, we plot on this chart the point ( s/ri), 
(S/N). If this point falls to the right of the diagonal 
line marked “No Decision Possible,” then it is more 
likely that Tactic 1 is preferable. How sure we are of 
this result, however, depends on the size of the quan- 
tity (Nri). The rule as given above can therefore be 
translated into the following: 

1. If the point (s/ri), (S/N) is to the right of the 
lower contour line corresponding to the product of 
the number of times Tactic 1 was tried and the num- 
ber of times Tactic 2 was tried, then use Tactic 1 
10 times as often as Tactic 2. 

2. If the point is between the lower contour for 
(Nri) and the diagonal line, use Tactic 1 twice as 
often as Tactic 2. 

3. If the point is to the left of the diagonal line, but 
is to the right of the upper contour for (nN), use 
Tactic 1 half as often as Tactic 2. 

4. If the point is to the left of the upper contour for 
(nN), use Tactic 1 one-tenth as often as Tactic 2. 

5. If situation (1) or (4) continues for several 
months, and if other intelligence indicates that it is 
likely to continue, discontinue Tactic 1 or Tactic 2 
entirely. 

Thus we have derived a set of rules which tell us 
what to do about introducing any particular counter- 
measure when the operation consists of discrete tries. 

One notices that the greater the product (nN) is, 
the sharper can be the distinction between courses of 
action. This corresponds to the general principle of 
probability, that a large number of trials reduces the 
chance of a misleading result due to fluctuations. This 
condition is sometimes difficult to achieve in prac- 
tice, for the enemy is changing his tactics, and it 
would not do to lump together data taken before and 
after a change in enemy tactics. From this point of 
view, it is preferable to use data taken over as short a 
period of time as possible, which means a small value 
for n and N. General rules cannot be made up for 
handling such situations; each case must be dealt 
with on its own merits, utilizing all available infor- 
mation, and exercising common sense. 


5.3.10 Continuous Operations 

The second situation, where a continual effort must 
be made, is dealt with in an analogous manner. As an 
example, we can consider the search for U-boats by 
antisubmarine aircraft. Tactic 1 will be the search by 
radar planes, and Tactic 2 can be the search by 


visual means with the radar set shut off. As long as 
the enemy does not use radar warning receivers 
effectively, the radar planes will discover more U- 
boats per thousand hours of flying than will visual 
planes. This situation, however, mil change when the 
warning receivers begin to get effective. 

Let us suppose that during the last month radar 
planes have flown e hours, searching for U-boats. 
During the same time and over the same portion of 
ocean, we suppose that nonradar planes have flown 
E hours. During this time, the radar planes have 
made m contacts with U-boats and the visual planes 
have made M contacts. If the effectiveness of radar 
planes, in contacts per hour, is p, and if the effec- 
tiveness of the visual planes is P, then the proba- 
bility that the number of contacts mentioned is 
actually obtained turns out to be : 


(pe) m (PE) M 

m\M\ 


• exp (-pe - PE) = f e (p, P ) 


from equation (28) of Chapter 2. 

If m and M are large enough, this expression may 
be approximated by one which is similar to the dis- 
crete case discussed above. 


Mp,p) 




e 2 E 2 


4:ir 2 mM 


r_ ( 

mV 

- - 1 

V- M Y1 

L 2m V 


2M 

\ e) J 


Returning now to Figure 6, we see that this function 
has its maxima at p = (m/e), P = (M/E). An argu- 
ment entirely analogous to that given for equation 
(12) shows that the probability that the effective- 
ness of Tactic 1 is greater than the effectiveness of 
Tactic 2 is given by the following expression. 

where F n is the function defined in Chapter 2, equa- 
tion (24). 

By arguments analogous to the discrete case dis- 
cussed above we can devise a new contour chart 
which will guide us in our decisions in the continuous 
case. This chart is given in Figure 8. The results of 
our test operations are expressed in terms of the 
position of the point (e/E), (n/N). Rules similar to 
those discussed above for the discrete case with 


CONFIDENTIAL 


102 


TACTICAL ANALYSIS 



RATIO EFFORT EXPENDED ON TACTIC \ TO EFFORT EXPENDED ON TACTIC 2,( e / E ) 

Figure 8. Continuous effort e and E expended on Tactics 1 and 2, resulting in m and M successes. Rules for future 
action, based on graph, are given in text. 


Tactic 1 and 2 reversed, apply here. We notice again 
that it is important to get as many contacts per 
month as possible, for the enemy is likely to change 
his tactics. 

5 4 THEORETICAL ANALYSIS OF 
COUNTERMEASURE ACTION 

The previous subsection considered the case where 
we were not sure how effective the enemy counter- 
measure was, nor how often he was using it. It also 
assumed that the enemy’s action was slow, so that 
data taken over a month would represent a particu- 


lar situation, which could be relied on to hold for 
another month or so. In other cases, however, intelli- 
gence is more complete and both sides know reason- 
ably well what the other side can do, and what value 
the choices have. Suppose both sides are keeping 
watch over the results of each action, and can change 
from one tactic to another as rapidly as they see 
what tactic the opponent chooses, if they can gain 
by change. 

In this case it pays to analyze the play in advance. 
The enemy has the choice of using a measure or not 
using it. Correspondingly, our own forces have the 
choice of using a countermeasure or not using it. This 


CONFIDENTIAL 



THEORETICAL ANALYSIS OF COUNTERMEASURE ACTION 


103 


makes four possible combinations, each of which have 
a certain value to our side and a corresponding dam- 
aging effect to the opposite side. This can be illus- 
trated in the following diagram : 


Wn, Value to our side, or damage to enemy. 




Enemy action 

Our 

action 


No use of 
measure 

Use of 
measure 

No use of counter- 
measure 

Wn 

Wn 


Use of counter- 
measure 

Wn 

w 22 


Sometimes W 2 1 is smaller than W n , indicating that 
our use of the countermeasure is a detriment to us if 
the enemy is not using the measure. On the other 
hand, Wi 2 is usually much smaller than W 22 if the 
countermeasure is at all effective, and W 12 is smaller 
than Wn if the measure is to profit the enemy (a 
small W is best for the enemy). As soon as we know 
the tactics of the enemy, we must adjust our own 
tactics so that the value Wij is as large as possible. 
Correspondingly, the enemy will try to adjust his 
tactics so as to make W as small as possible. This is an 
example of a situation which also occurs in many 
games. Its enunciation is usually called the minimax 
principle . 4 One side wishes to maximize W, and the 
other wishes to minimize it. This has been discussed 
in connection with equation ( 16 ) of Chapter 4 . 

5.4.1 Definite Case 

There are a number of possibilities, which may be 
analyzed separately. The first, which we can call the 
definite case , occurs when W 2 \ is larger than Wn (we 
have already assumed that W 22 > W 12 and Wn > 
Wvi). In this case we should always use the counter- 
measure, for, no matter what the enemy does, we 
would profit by its use (W 2 \ > W n and W 22 > W 12 ). 
The enemy, if his intelligence is good, would know 
this, and would choose always to use the measure if 
W 22 < W 2h or not to use it if W21 < W 22 . The point 
here is that, if both sides know the values of all the 
combinations, they will always prefer the one tactic 
which will give them the most gain (or the least loss) 
no matter what the other does. A similar case occurs 
when Wn > W 2 \, but W 2 1 > W 22 , for here it is always 
to the advantage for the enemy to use his measure. 


5.4.2 Indefinite Case 

The other case, which will be called the indefinite 
case , for reasons which will shortly become clear, is 
where Wn > W 2 \ and IT22 > W 2 i (we lose by using 
the countermeasure if the enemy is not using the 
measure, and the enemy loses by using the measure 
if we are using the countermeasure). We can see the 
difficulty if we try to figure out what we should do if 
the enemy’s espionage were perfect, and compare it 
with what the enemy should do if our espionage were 
perfect. (To make the analysis specific, let us take 
IT22 > Wn > W 2 i > Wn.) In the first case, if we 
should decide to use the countermeasure, the enemy, 
knowing our decision in advance, would use no 
measure and we would get W 2 i; if we should decide 
not to use the countermeasure, the enemy would use 
his measure, and we would get Wi 2 . Since W 2 i > Wi 2 , 
we would prefer to use countermeasures in this case, 
since IT21 is the maximin of the array (i.e., in each 
row there is a minimum value; W21 is the largest of 
these). On the other hand, if we knew beforehand 
that the enemy was going to use his measure, we cer- 
tainly would use our countermeasure, getting IT22; or, 
if we knew the enemy were not going to use his meas- 
ure, we would not use our countermeasure, and the 
result would be W n - In such a case, it would behoove 
the enemy not to use his measure, for Wn < W 22 ; Wn 
being the minimax of the array (i.e., in each column 
there is a maximum value; Wn is the least of these). 

The property of the definite case, which makes it 
definite, is that the minimax is the same as the maxi- 
min, so that there is no question as to which each 
side should do. In the indefinite case, however, the 
minimax is not the same as the maximin, so that it is 
not obvious what either side should do. The correct 
solution 4 is that neither side should stick to one tac- 
tic, but should alternate them at random, with some 
predetermined relative emphasis between the two 
choices. The difficult part of the problem is the de- 
termination of this emphasis. 

5.4.3 Solution for the Indefinite Case 

To make the problem specific, let us choose values 


for the IT’s: 

They 

No 

measure Measure 

We No CM 

2 

0 

CM 

1 

3 


CONFIDENTIAL 


104 


TACTICAL ANALYSIS 



la 

Figure 9. Choice 


Now suppose the enemy varied his tactics, some- 
times using the measure and other times not using it. 
We could plot the average return to our side as a 
function of the percentage of times the enemy used 
his measure; one curve for when we used the counter- 
measure, and another for when we did not. These are 
shown in Figure 9A. We see that the safest mixture 
for the enemy is to use the measure one-quarter of 
the time (point P). For if he tries any other mixture 



B. FRACTION OF TIMES WE USE COUNTERMEASURE 


Zb 



D 


of mixed tactics. 

we can get better results; for instance, if he uses his 
measure 50 per cent of the time, we can get best 
results by using our countermeasure all of the time, 
and, if he uses his measure only 10 per cent of the 
time, we can win by not using our countermeasure. 
His mixture of tactics, moreover, must be made 
completely at random, otherwise we might be able to 
gain by following his pattern of choice. The average 
value to us is 1.5, if he uses the safest mixture. 


CONFIDENTIAL 


THEORETICAL ANALYSIS OF COUNTERMEASURE ACTION 


105 


We also must protect ourselves by using a random 
pattern of tactics, as is shown in Figure 9B. Unless we 
use equal amounts of countermeasure and no counter- 
measure (point Q), the enemy may be able to gain an 
additional advantage. This mixture must also be dis- 
tributed at random. We must, of course, keep track 
of the enemy’s frequency of using his measure, so 
that if he drifts off point P (or if any of the W’s 
change value) we can adjust our tactics to gain 
by it. 


5.4.4 Case of Three Choices 


The problem above was concerned with two choices 
on the part of each adversary, but similar problems 
can occur with more choices. We can consider the 
possibility of two different measures (or none) on the 
part of the enemy, and two possible countermeasures 
(or none) on our part. This is indicated in the follow- 
ing table where Tactic 3 a might be no measure and 
Tactic 36 might be no countermeasure, with la and 
2 a two different counters. 


Tactic 16 
Side 6 Tactic 26 
Tactic 36 


Tactic 

Side a 
Tactic 

Tactic 

la 

2 a 

3 a 

4 

1 

1 

0 

3 

1 

0 

0 

2 


(Value 
to side 6) 


Here we must plot the results of mixed tactics on a 
triangular chart, where the distance in from each 
edge represents the percentage of use of one of the 
tactics, and the distance perpendicular to the plane 
of the triangle is the value to side 6. Such plots are 
shown in Figure 9C and D. At the edge of each side 
is a side view giving the traces of the planes represent- 
ing pure tactics for the other side. The shaded parts 
of the triangle give the best choice of tactic for any 
mixture of opponent’s tactics. We see that, if side a 
uses any mixture other than ^ of la, f of 2a, and f of 
3a, side 6 will be able to gain; if side 6 uses any other 
mixture than ^ of 16, -J of 26, and ^ of 36, side a will 
be able to make more. The net value to side 6, if 
both sides play “correctly,” is If side 6 tried to 
make more than this, an alert enemy could arrange 
it so side 6 would make less. Therefore, this mixed 
solution is the safest solution; presumably it should 
be used whenever we do not know what the enemy is 
liable to do, and should only be modified when we 
are reasonably sure the enemy has departed from his 
“safe” mixture. 


5.4.5 Barrier Patrol for Submarines 

There are cases with an infinite variety of choices 
of tactics and countertactics, where a mixture of tac- 
tics is necessary for safety. An example is in the 
(much simplified) problem of the barrier patrol of a 
plane guarding a channel from passage by a subma- 
rine. We suppose that the submarine cannot be 
caught by the plane when it is submerged, but that 
it can only run a distance a submerged. Each day 
the plane can fly back and forth across the channel in 
one part, though the next day it can patrol another 
part of the channel. If it patrols a wide part of the 
channel it cannot do it as efficiently, but, if it always 
patrols across the narrowest part, the submarines 
can dive and elude it. The position of patrol along 
the channel must be varied from day to day so as to 
keep the submarine from being certain. 

The situation is illustrated in Figure 10. A channel 
of length larger than a is to be patrolled by the planes. 
If the barrier patrol is at point x and if the submarine 
attempts to go by on the surface, the probability 
that the plane will contact the submarine is given by 
P(x) as shown in the lower half of Figure 10. 

As we have said above, the barrier cannot be 
placed always at the same point x; it must be placed 
here and there along the channel so that the subma- 
rine can never be sure exactly where it is placed. The 
extent of the range of values of x over which the 
barrier is placed must, of course, be longer than a; 
otherwise the submarine could dive under the whole 
distribution. The relative frequency of times the bar- 
rier is placed at a point x is proportional to the proba- 
bility density <£(x). Since the barrier is to be placed 
somewhere each day, the integral of </> over x must be 
unity. Side A must then adjust the shape of the curve 
0 so that it gets good results no matter what the sub- 
marine does. The submarine can also have a choice 
of its point of submergence. It will always run sub- 
merged its maximum distance a. It can, of course, 
come to the surface before a distance a, and resub- 
merge for the rest of a at some other region in the 
channel, if this seems best. There will be, therefore, 
a certain probability \J/(x) that the submarine is sub- 
merged at the point x. The integral of over x must 
equal a, if the submarine is to have a maximum sub- 
merged range of a. The submarine (Side B) must, 
therefore, adjust probability \p so that it gets as good 
a result as possible no matter where the barrier is 
placed. 


CONFIDENTIAL 


106 


TACTICAL ANALYSIS 




Figure 10. Barrier patrol versus diving submarine. 


This is indicated in the following equations : 


P(x) = Probability of contact if barrier is at 
x and submarine is surfaced. 

<t>(x) = Probability density for barrier at x. 



\//(x) = Probability that submarine is sub- 
merged at x.J^{x)dx = a . 

G = Probability of contact 

=f P{x)4>(x) [1 - \p{x)]dx. 


Side A adjusts the probability density </> so that 
the probability of contact G is as large as possible and 
is independent of the tactics of the submarine, within 
reason. Conversely, Side B, the submarine, adjusts 
the probability of submergence <j> so that G is as small 
as possible and is independent of the placing of the 
barrier, within reason. The problem is to determine 
the function 4> and \f/ for safe tactics for both sides 
(14) w hen the probability P is known. 

We will first determine the tactics of the subma- 
rine, determined by the function \p. Suppose the sub- 
marine has chosen a function \f/, and suppose by 
chance side A has found out what this function is. A 
glance at the equation for G shows that, if the quan- 
tity, P(#)[l — yj/{x)], has a maximum value, then 


CONFIDENTIAL 


THEORETICAL ANALYSIS OF COUNTERMEASURE ACTION 


107 


SUBMARINE TACTICS (SIDE B) 



BARRIER TACTICS (SIDE A) 



(h)( SHADED AREA) = | 




Figure 11. Solutions of problem shown in Figure 10 for safe tactics for both sides. It turns out that H = K, and that 
the probability for contact G is equal to H, which equals K = h(L — a). Function <j>(x) controls the frequency of placing 
barrier at point x, and \p(x) is optimum probability that submarine is submerged at x. 


Side A will get the best results by placing its barrier 
always at the x corresponding to this maximum 
value. The submarine would like to make this quan- 
tity zero, but this is not possible, since the channel is 
longer than the maximum range of submergence, a. 
In any case, however, the submarines can choose xf/ so 
that the quantity P( 1 — \p) has no single maximum ; 
i.e., is flat along the top. This is done as follows : 

Safe tactics for Side B ( submarine ). 

Over as great a range of x as possible, choose 
\J/ so that 

P(x) [ 1 — \p(x) ] = H, a constant, i.e., 

1 ~ 1p{x) ’ when p ^ H ’ ^ 

|0, when P(x) < H . 


H is determined by the condition : 

/( i_ f) & = a - 

Probability of contact :G ^ H. 

The integral determining H ensures that H is as 
small as possible. If now the barrier is placed any- 
where in the range of x where the submarine sub- 
merges, the probability of catching the submarine 
will be H ; if the barrier is placed in a position where \p 
is zero (where the submarine does not submerge) 
then we have adjusted things so that the probability 
of contact G is smaller than H. Consequently, this 
distribution of diving is the best the submarine can 
manage. 

The details of the solution are indicated in the two 
drawings of Figure 11 representing the submarine 


CONFIDENTIAL 


108 


TACTICAL ANALYSIS 


tactics. We plot the reciprocal of the probability P 
and draw a horizontal dashed line of height (1 /H) so 
adjusted that the area between this dashed line and 
the (1/P) curve is equal to ( a/H ). When this is done, 
the probability of submergence will be proportional 
to the difference between the horizontal dashed line 
and the (1/P) curve, as shown in the second drawing. 
The probability of contact is therefore equal to H if 
the barrier is anywhere in the region between Xq and 
Xij and is less than H if the barrier patrol is placed 
outside of this range. If a is small, it may turn out 
that the horizontal dashed line must be so low as to 
break the shaded area into two parts (for the type 
of P shown). In such a case, it is best for the subma- 
rine to submerge in two separated regions, since it 
must conserve its short range of submergence for 
those regions where the probability of detection is 
highest. 

We must now see what the side controlling the 
barrier (Side A) is to do about its choice of the den- 
sity function 0. Suppose it chooses a particular func- 
tion 0, and suppose the submarine learns what its 
values are. The submarine will then always submerge 
in those regions where P(x)<t>(x) is largest. As with 
the submarine, therefore, the barrier patrol plans 
must be made so as not to have any maxima for the 
function P<£. Consequently, no barrier will be flown 
at positions where the probability P is less than some 
limiting value K; and, where P is larger than K, the 
barrier density will be inversely proportional to P, in 
order that the product P<f> will be constant in these 
regions. The equations corresponding to this require- 
ment are 

Safe tactics for Side A (barrier). 


The problem for the distribution of the barrier is 
not yet completely solved, since we can vary the 
value of L and readjust the value of h. It certainly 
would not be advisable to make L smaller than a, 
for then the submarine could dive completely under 
the system of barriers. On the other hand, it would 
not do to make L too large, for then h would become 
too small. In fact, we must adjust h and L so that 
the quantity h(L — a) is a maximum. 

A certain amount of algebraic juggling must be 
gone through to show that the requirements that 
h(L — a) is a maximum correspond to the require- 
ments that K — PL. In other words, the range of x 
which is covered by the barrier turns out to be the 
same range which is covered by the submerging sub- 
marine. This is, of course, reasonable, for any overlap 
on the part of either side would represent waste mo- 
tion and reduced effectiveness. The technique for 
obtaining the distribution for barrier patrol </> is 
shown in the right-hand curves in Figure 11. One 
again plots the reciprocal of P(x), chooses K equal 
to H and the end point x 0 and x\ to be the same as in 
the analysis for the submarine. We choose the nor- 
malizing factor h to be such that h times the area 
shaded in the right-hand figure equals unity. There- 
fore, the best the submarine can do, if the barrier 
patrol is fully alert, and the best the barrier patrol 
can do, if the submarine commander is clever, is the 
set of tactics defined by equations (15) and (16), with 
the resulting probability of contact given below : 

Safe tactics for both sides 

Integration is carried out over that region 
of x where P(x) > H. The constant H is re- 
lated to the maximum submerged range a by 


Choose <h(x) so that P(f> is constant over a range 
of x, i.e., 


^) = (/ 4 ) ,whenP(x)>x ’ 

10, when P(x) < K , 


( 16 ) 


where the range over which P > K is between 
Xq and x\(xi — Xo = L) . 


The value of h is determined by the condition : 


H = h(L — a), where- = f 

h J P(x) 


and L = fdx is the total range of x over 
which P is greater than H. The probability 
of submergence for the submarine is then 
given in equation (15); the barrier density 
is 


f — * , 

4>(p) (L - a)P{x) 

[ 0 , 


P(x) >H, 
P{x) < H; 


( 17 ) 



and the probability of contact of plane and 
submarine is 


G — H = h(L — a) 


L — a 



Probability of contact: G ^ h(L — a), which 
is maximum when K = H. 


CONFIDENTIAL 


THEORETICAL ANALYSIS OF COUNTERMEASURE ACTION 


109 


A number of other examples might be considered, 
where this method of analysis of tactic and counter- 
tactic would be useful. The difficulties at present are 
in finding a general technique for the solution of such 
problems. The studies of Von Neumann and Morgen- 
stern 4 show that there are solutions to each problem 
and show the general nature of these solutions. They 
do not show, however, the technique for obtaining a 


solution. We have seen examples of the problems and 
how their solutions can be obtained in three cases. A 
great deal more work needs to be done in finding so- 
lutions to various examples before we can say that 
we know the subject thoroughly. 

It is to be hoped that further mathematical re- 
search can be undertaken on this interesting and fruit- 
ful subject. 


CONFIDENTIAL 


Chapter 6 

GUNNERY AND BOMBARDMENT PROBLEMS 


I n this chapter we shall consider a class of prob- 
lems which arise in the evaluation of such weap- 
ons as guns, bombs, and torpedoes, and in analyzing 
the best methods for their use. All these weapons are 
used for the destruction of targets, such as ships, gun 
emplacements, factories, and the like. The effective- 
ness of any such weapon against a given kind of 
target can be measured, at least in part, by the ratio 
of the number of targets destroyed to the number of 
shells, bombs, or torpedoes used. 

The majority of this sort of analysis should, of 
course, be carried out by scientists in the service lab- 
oratories, where the weapons are designed or pro- 
duced. However, the operations research worker 
should be familiar with the techniques of evaluating 
weapon performance, and should know enough about 
the behavior of the weapon to be able to analyze its 
action if necessary. The operations research worker 
often is useful as a technical adviser to the users of 
equipment giving unbiased advice on technical details 
of use, although his primary task is the broader one of 
helping fit equipment, personnel, training, intelli- 
gence, etc., into an effective operation. This chapter 
is included, as was Chapter 2, because it is believed 
that ability in the broader problems is improved by 
mastery of the narrower technical details. 

The number of shells, bombs, or torpedoes re- 
quired, on the average, to destroy a target depends 
on two primary factors: the destructiveness of the 
weapon, i.e., the probability that the target is de- 
stroyed if the weapon hits it, and the accuracy of the 
weapon, i.e., the probability that the weapon will hit 
the target. In addition, if weapons are used in groups 
rather than singly (for example, spreads of torpedoes, 
sticks of bombs) the result depends on the firing 
pattern used. In this chapter we shall describe 
methods of calculating the probabilities of destroy- 
ing targets, and of determining the patterns which 
create the maximum destruction. 

61 THE DESTRUCTIVENESS OF 
WEAPONS-LETHAL AREA 

The simplest case of measurement of destructive- 
ness occurs when the weapon must hit the target in 
order to destroy it, but always does destroy the tar- 
get if it hits. Such a case is found, for example, in use 


of medium-caliber (e.g., 5-inch) shells against open 
gun emplacements. The walls of such emplacements 
are, in general, strong enough to protect the guns 
and men within them against the blast and frag- 
ments from shells which strike outside the emplace- 
ment, whereas, if such a shell hits inside the em- 
placement, both guns and men are destroyed. The 
probability that a shell destroy the emplacement is 
therefore just the probability that the shell hits a 
certain area, called the lethal area. The magnitude of 
the lethal area is a measure of the destructiveness of 
the shell against these targets. 

As a slightly more complicated example, let us con- 
sider some cases of antisubmarine ordnance. The 
simplest case is that of a small contact charge 
(hedgehog or mousetrap). This situation is very 
similar to the last : the charge must hit the submarine 
to destroy it, and it is usually assumed that a single 
hit is sufficient to cripple the submarine. In this case 
the lethal area is the area of the horizontal cross 
section of the submarine or, better, of its pressure 
hull. 

If we consider the case of a depth charge with a 
proximity (or influence) fuze, it is no longer neces- 
sary actually to hit the submarine to destroy it, for 
the proximity fuze will convert a near miss into a 
destructive explosion. We let R be the radius of 
action of the proximity fuze. If the charge is sufficient 
so that an explosion within this radius from the sub- 
marine will sink the submarine, then the lethal area 
is increased to the area included within a curve sur- 
rounding the horizontal cross section of the submarine, 
at a distance R from its boundary. Finally, if we con- 
sider the case of the depth charge set to explode at a 
preset depth, we no longer have a lethal area at all, 
but a lethal volume, for, to sink the submarine, the 
depth charge must explode within the volume en- 
closed by a surface surrounding the submarine at a 
distance equal to the lethal radius of the depth 
charges. 

6.11 Multiple Hits 

In many cases a single hit is not enough to destroy 
a target. A typical case is that of a torpedo hitting a 
ship. Even for merchant ships a single torpedo hit is 
not usually enough to sink it, and heavy combatant 


110 


CONFIDENTIAL 


THE DESTRUCTIVENESS OF WEAPONS-LETHAL AREA 


111 


ships are designed to withstand many torpedo hits. 
To treat such cases exactly, we should determine the 
probabilities D h D 2 , D 3 , etc., of sinking the ship if 
the ship is hit 1, 2, 3, etc., times. Then, if, for a given 
method of firing torpedoes, the probabilities of 1, 2, 
3, etc., hits are Pi, P 2 , Pz, etc., the probability of 
sinking the ship would be 

Pa — DiP 1 + D 2 P 2 + D 3 P 3 + • • • ( 1 ) 

The probabilities Pi, P 2 , P 3, etc., are sometimes called 
the damage coefficients. 

As a matter of operational experience it has been 
found that the damage coefficients in many cases are 
related, to a very good approximation, by the law 

P n =l-(1-Pi)", (2) 

which is just the law of composition of independent 
probabilities (see Chapter 2). In other words, the 
chance of sinking a ship with any torpedo hit is 
always the same, regardless of how many previous 
hits there have been. This may be interpreted as 
meaning that a torpedo will only sink a ship if it 
hits a “vital spot,” and that hits on other than vital 
spots will damage, but not sink, the ship. This “vital 
spot” hypothesis, while not to be taken too seriously 
as an actual description of what happens when ships 
are hit by torpedoes, does serve to reduce the number 
of unknown quantities P n to one, and has been found 
to give fairly satisfactory results in many cases, not 
only for sinking cargo ships by torpedoes, but also in 
many other cases, such as AA hits on aircraft. How- 
ever, it is not a very good approximation for battle- 
ships, with many compartments. 

When the “vital spot” theory can be applied, the 
lethal area of a target is defined as the product of the 
effective area of the target and the probability D 
(the same as our previous Hi) that a hit on this area 
will destroy the target. For example, a torpedo hit 
on a merchant vessel has a probability of about 1/3 
sinking the ship. Hence D = 1/3, and the lethal area 
is 1/3 of the length of the ship. 

In some problems it is necessary to take into 
account the variation of the probability of destroy- 
ing the target as a function of the point at which the 
hit is made. If we consider, for example, the effect of 
proximity-fuzed AA shells on aircraft, the probability 
of destruction is high if a shell hits or passes very 
close to the aircraft. As the trajectory moves further 


away from the aircraft the probability of destruction 
decreases until finally a distance is reached at which 
the fuze is no longer activated, and the chance of 
destruction falls to zero. In such cases we must 
express this probability as a function D{x,y), the 
damage function, where x and y are coordinates cen- 
tered on the target in a plane perpendicular to the 
trajectories. When this has been done, the lethal 
area may be defined as 

'-If D{x, y)dxdy , (3) 

where the integration is over all the area for which 
D(x, y) is greater than zero. It is easy to see that the 
simpler definitions of lethal area are included in (3) 
as special cases in which D(x, y) has a constant value 
over the target area. The modification of (3) to cases 
of one- or three-dimensional targets is obvious. 

6-1-2 Random or Area Bombardment 

In cases where bombs or shells are dropped at 
random over an area, the lethal area is sufficient to 
determine the destruction which will be caused. For 
any given target in the area, the chance that a given 
bomb or shell will hit the element of area dxdy is 
just {dxdy/ A), where A is the total area bombarded. 
The probability that this bomb or shell will destroy 
the target is therefore 

P 1= f D(x,y) d -^, (4) 

or 

P^ = - A . ( 5 ) 

A 

If n bombs or shells are dropped, the probability 
that a given target is destroyed is 

P n = 1 - e~ nL/A . (6) 

This is not only the probability that one particular 
target is destroyed, but also represents the expected 
fraction of all the targets in the area which are de- 
stroyed. The result is a generalization of that given 
in Section 5.2. 

As a numerical example, let us suppose that an area 
of 1 square (nautical) mile is to be bombed with 1,000- 
pound GP bombs. The area contains 100 gun em- 


CONFIDENTIAL 


112 


GUNNERY AND BOMBARDMENT PROBLEMS 


placements (each of lethal area 400 square feet) and 
personnel in trenches [total lethal area per man, de- 
termined by equation (3) of 900 square feet]. Since 
1 square mile is 36,000,000 square feet, the ratio L/A 
is 1/90,000 for the gun emplacements, and 1/40,000 
for the personnel. A plot of the fraction of the gun 
emplacements and personnel destroyed, as a func- 
tion of n, is shown in Figure 1. It will be noted that 


< i 

z § 


z o 



i 

cr 



Figure 1. Destruction by area bombardment. 


an enormous expenditure of ammunition is required 
to accomplish much destruction by area bombard- 
ment. 


6-1-3 Aimed F ire— Small T argets 

We now consider the case in which the bombs or 
shells are not distributed at random over an area, 
but are each individually aimed at a target. For the 
present, however, we shall suppose that the target 
dimensions are small compared to the aiming errors. 
When this is the case, we may neglect the variation 
over the target area in the probability of hitting an 
area element dxdy, and again the lethal area is 
sufficient to determine the destructiveness. 

If the bombing errors follow the normal distribu- 
tion law (usually a safe assumption), with a standard 
deviation in range o r , and a standard deviation in 
deflection a d , then the probability of hitting the area 
element dxdy at x, y, where x is the range error and 
y the deflection error, is 

P(x, y)dxdy = -F_ e -WW-WM) dxdy ( 7) 

2l r(T r (7d 

Near the target this has the value 

P{ 0, 0)dxdy = . 

2Tra r (Td 


The probability of destroying the target with a single 
bomb or shell is 



D(x, y)dxdy 
2lT(J r O d 


L 

2 7 T O r O d 


( 8 ) 


If n bombs or shells are dropped independently, the 
probability of destruction is 



Since by hypothesis L is small compared to o r o d , this 
is approximately 

P n = 1 - e “ nI ' /Wd . (9) 

To compare this result with the case of area bom- 
bardment, let us consider that the square mile of our 
last example contained 100 gun emplacements (each 
of lethal area 400 square feet), and that the standard 
deviations of the bombing errors, o r and o d , are each 
200 feet. Then the ratio L/2ivo r od = 400/(27r)(200) 2 
= 1/ 600 approximately. The number of bombs which 
we must expect to drop to destroy all 100 emplace- 
ments would be 60,000 bombs. It should be noted 
that this is the number which would be required if 
each target was bombed until it was destroyed. If it 
were arbitrarily decided to drop 600 bombs on each 
gun emplacement, then by equation (9) the fraction 
destroyed would be (1 — e -1 ) or 0.63, a much poorer 
showing. 


6.1.4 Aimed Fire— Large Targets 

When the assumption can no longer be made that 
the target area is small compared to the aiming errors, 
it becomes necessary to consider the variation of the 
probability of hitting an area element dxdy over the 
target area, and the idea of lethal area loses most of 
its usefulness. If f(x, y)dxdy is the probability of hit- 
ting an area element, the chance that the target is 
destroyed by a single shot becomes 

-Sf D(x,y)f(x,y)dxdy , (10) 


and further progress depends on our ability to evalu- 
ate this integral. Once evaluated, however, we have, 
as before, the result that the expected number of 


CONFIDENTIAL 


PATTERN FIRING-NO BALLISTIC DISPERSION 


113 


shots to destroy the target is 1/Pi, and the proba- 
bility of destroying the target in n shots is 

P n =l-(l-Pi)\ (11) 

which, if Pi is small, can be written 

P n = 1 - e -nPl . (121 

If D(x,y) has a constant value D over the target 
area, then equation (10) represents just the chance 
of hitting the target, multiplied by D. If the target 
is of simple shape, and the probability density f(x, y ) 
is not too complicated, then the evaluation can be 
carried out. As an example of this type of calculation, 
consider the following problem: 

A submarine fires torpedoes at a merchant ship 
200 feet long, and at a track angle of 90 degrees. The 
errors in firing are normally distributed with a 
standard error of 100 mils (somewhat larger than the 
errors experienced in World War II, but taken here 
as an example), its shots are fired from 2,000 yards, 
and the probability of sinking due to a torpedo hit 
is 1/3. We wish to know the chance of sinking the 
ship with n shots, each fired independently. 

The standard deviation of the torpedoes in distance 
along the target’s track is 100/1,000 X 2,000 = 200 
yards or 600 feet. Hence the probability of an error 
of between x and x + dx feet measured from the 
center of the target is 


The expected number of torpedoes required to sink 
the ship is 23. 

As a second example, let us consider the gun em- 
placements we have considered before. Let us con- 
sider them to be circular, with a radius of 11 feet, 
and let D(x, y) = 1 inside this radius, and 0 outside. 
We shall suppose that the standard deviations of the 
errors in range and deflection are now 50 feet, in- 
stead of the previous values. Then 

4>(x, y)dxdy = — - — e~ ix * +y2)/2( - 50) * dxdy 
2tt(50) 2 

= — l—e-^MM'r-drde. 

2ir(50) 2 


where r and 6 are the usual plane polar coordinates. 
Then 


P 1= r7“_i_ e - 

Jo Jo 2tt(50) 2 


2 /2(50) 2 


rdrdO 




= 0.0239 . 

If this is compared with the small-target result 
= = o 0 242 

2tt(50) 2 


1 

- 7 = exp 

V 2tt • 600 



— —\dx. 
2(600) 2 / 


we see that the agreement is very good. If, on the 
other hand, the radius had been 50 feet, then 


The probability of hitting the ship is therefore 
100 ! 


/ 


-100 V 2jr . 600 


e — (XV2 - 600 


=_L f 


1/6 


/ e~ W2) dy = 0.132. 


1/6 



while the small-target approximation would have 
given 


p _ tt(50) 2 
1 2tt(50) 2 


0.50. 


The chance of sinking the ship is 1/3 of this or 0.044. 
The probability of sinking the ship with 1, 2, • • • 6 
torpedoes is given in the following table : 

No. torpedoes Prob. of sinking 

1 0.044 

2 0.086 

3 0.126 

4 0.165 

5 0.202 

6 0.237 


In this case the small-target approximation is not 
satisfactory. 

6.2 PATTERN FIRING-NO BALLISTIC 
DISPERSION 

There are many tactical situations where it is ad- 
vantageous to fire several shots (or torpedoes, or 
bombs) more or less simultaneously, instead of aim- 
ing each shot individually and firing consecutively. 


CONFIDENTIAL 


114 


GUNNERY AND BOMBARDMENT PROBLEMS 


They can all be fired in the same direction ( salvo 
firing) , or each shot can be displaced a predetermined 
amount with respect to the others ( pattern firing ) ; 
the relative advantages of the two methods are de- 
termined by the errors involved. There is, first, the 
error in the aiming of the salvo or the center of the 
pattern; this is called the aiming error. Secondly, 
there is the spread of the individual shots as they 
travel toward the target, converting the salvo into 
an irregular pattern, and changing a regular pattern 
into an irregular one; this is called the ballistic error. 

If the dispersion of the ballistic errors is larger than 
the dispersion of the aiming errors and also larger 
than the lethal area of the target, the best one can do 
is to fire a salvo (zero pattern spread). On the other 
hand, if the aiming dispersion is larger than the 
ballistic dispersion and also larger than the lethal 
dimensions of the target, it is usually better to use 
pattern firing. When salvos are fired, if the ballistic 
errors are very small, the shots will either all hit or 
none will hit; whereas, if the shots are spread into a 
pattern, it will be more likely that at least one shot 
hits. This is the shotgun method, as opposed to the 
rifle. Since this case often occurs in practice, it is 
important to study methods for determining opti- 
mum pattern shape and size. 

To bring out the fundamental principles involved, 
we shall first consider that the ballistic dispersion is 
negligible compared to the aiming dispersion. If a 
regular pattern is fired, a regular pattern will arrive 
near the target. We shall soon see that, if the pattern 
is too small, either several shots will hit or they will 
all miss; as the pattern size is increased, the chance 
of at least one hit increases to a maximum, represent- 
ing the optimum pattern. For still larger patterns the 
probability of at least one hit decreases again. This is 
the basic philosophy of dropping bombs in sticks or 
firing torpedoes in spreads. 

6.2.1 Example from Train Bombing 

The fundamental problem to be solved in pattern 
firing is that of finding the best pattern to use. To 
illustrate, let us consider the following simple case. 
A plane carries two bombs to attack a single-track 
railroad. A single bomb hit within 25 feet of the 
center of the track is sufficient to destroy the track. 
The plane therefore flies along a course perpendicu- 
lar to the track, and drops a pattern of two bombs, 
spaced a distance 2 a apart, aiming the midpoint of 
the pattern at the center of the track. We wish to 
determine the best stick spacing. 


If the error of aiming the pattern is x, i.e., if the 
midpoint of the pattern falls a distance x beyond the 
track, then the track is destroyed if — 25 < ( — a + x) 
< 25, or if — 25 < (a + x) < 25. We may therefore 
introduce the 'pattern damage function, D p (x), the 
probability that the target is destroyed if the center 
of the pattern falls at the point x , which is given, in 
this case, by 

f I, if — 25 < (— a + x) < 25 , or 
D p (x) = <! -25 < (o + x) < 25. 

[ 0, otherwise. 

If f p (x)(dx) is the probability that the center of the 
pattern falls in the element dx, then the probability 
of destroying the target is 

P(a) = J D p (x)f p (x)(dx) . (13) 


This equation is completely analogous to equation 
(10). It will be noted, however, that the pattern 
spacing enters this as a parameter. The best pattern 
is that which makes P(a) a maximum. 

In the present case, equation (13) reduces to the 
form 


P(a) = 


/ a + 25 

f p (x)(dx) 

-a -25 


(a < 25) . 
(14) 


/ -a +25 ra+ 25 

f p {x)dx + / <j> p (x)dx (a >25) 

-a —25 J a— 25 


In every practical case the aiming error has a normal 
distribution, so that, if a is the standard deviation of 
the aiming error, 


Then 


/,(*) = 

V27TCT 


P(a) = 


r i 

ra+ 25 

/ e-^dx 

' -a -25 


a/ 2tt a 


2 

a/ 2tt ^ 

r (o+25) h 

/ e-^di, 

f 0 

(< a < 25) 

1 i 

r -o+25 

( e-^dx 

-a -25 

(15) 

a/ 2tt a J 


+ 


ra +25 
2tt dJ a -25 


-xV2<7 2 


a/ 27 r 

= — f 


dx 


(a +25)/<r 


e" |2/2 d! (a >25). 


(a -25) A 


In the range 0 < a < 25, P(a) is obviously increas- 
ing, while in the range 25 < a < °° f P(a) is decreas- 
ing. P(a) is therefore maximum for a = 25. Hence the 


CONFIDENTIAL 


PATTERN FIRING-NO BALLISTIC DISPERSION 


115 


a 



Figure 2. Damage function for spread of three torpedoes. No ballistic dispersion. D p = 19/27 in triple-shaded 
region, 5/9 in double-shaded region, 1/3 in single-shaded region, 0 in unshaded region. 


best stick spacing is 2a = 50 feet. This is obvious 
without calculation for the present simple case, but 
in many cases the analysis is necessary to obtain the 
correct answer. 

It is of some interest to compare the chance of de- 
struction in the three cases: (1) two bombs dropped 
together in salvo, (2) two bombs spaced 50 feet, (3) 
two bombs dropped on separate runs over the target. 
The chances of destruction in the three cases are 

o /*25/<r 

Case (1) P a = -== I e~ w di; 

V 2x*' 0 

o r50/a 

Case (2) P b = -== / e~ w dk) 

V27r^° 

Case (3) P c = 1 - (1 - P a ) 2 = 2 P a - P * . 

The values of these probabilities, for a number of 
values of a, are shown in the following table. 


Table of probability of destroying railroad track 
(at least one hit) 

P a = probability with 2 bombs in salvo, 

Pb = probability with 2 bombs, 50 feet spacing, 
P c = probability with 2 independent bombs. 

Standard 
deviation of 
bombing 
error 


M 

Pa 

P b 

Pc 

10 ft 

0.9876 

1.0000 

0.9998 

20 

0.7888 

0.9876 

0.9554 

30 

0.5934 

0.9050 

0.8347 

40 

0.4680 

0.7888 

0.7170 

50 

0.3830 

0.6826 

0.6193 

60 

0.3256 

0.5934 

0.5452 

70 

0.3812 

0.5222 

0.4833 

80 

0.2434 

0.4680 

0.4276 

90 

0.2206 

0.4246 

0.3925 

100 

0.1974 

0.3830 

0.3558 


CONFIDENTIAL 


116 


GUNNERY AND BOMBARDMENT PROBLEMS 


One sees that as long as ballistic errors are negligible, 
and as long as only one hit is needed for destruction, 
the salvo is never as effective as the pattern or as 
independent bombs. 

As a second simple example, suppose that a sub- 
marine, knowing it can fire only one spread of three 
torpedoes at a ship, wishes to obtain the greatest 
probability of sinking the ship. We shall suppose that 
the torpedoes run perfectly true, that the center tor- 
pedo is aimed at the center of the target, and the 
other two are equally spaced on either side. We shall 
assume the vital spot hypothesis with Di = 1/3, so 
that with one hit the chance of sinking the ship is 1/3 ; 
with two hits, 5/9; and with three hits, 19/27. Let 
21 be the length of the ship, and a the distance apart, 
at the ship, of adjacent torpedoes in the spread. Let x 
be the aiming error of the spread. The damage func- 
tion for any value of x and a will be that shown in 
Figure 2 (the three bands in the figure are the regions 
in which each of the torpedoes hit) . Applying this to 
equation (13), and assuming normal aiming errors, 
we find 


P(a) = 


2 f 

V2t 


(1-0) I- 




-* n du 


+ - -L / e-“ V2 

9 V2t 

1 <2 /~(l+a)/<r 


du 


i 2 r { 
3 Vzn- J i/« 
i 2 r 
3 V&r J 0 


3 V2tt J i/c 

'( a-l)/c 

•l/a 


e~ u2/2 du, 
e~ ui/2 dn 


i 5 2 . 

H — - 7— / e 

9 V2t 


- m */2 


du 


+ 


- — f 

3 '\Z2tt J V* 

i 2 J T 


(a+l)/* 


-u 2 /2 


3 \/ 2tt 

i _2_ r (o 
3 "\/ 9-jt J (° — 


e 

(a+D/a- 


du 


+ - -== / e-* /2 du, (2 l<a) . 

3 V 2tt J (a-l)/* 


The curves of P(a) against a for a number of values 
of <7 are shown in Figure 3. The curve of the optimum 
value of a as a function of a is shown in Figure 4. 

6.2.2 The Squid Problem 

As a somewhat more complicated example, we 
shall now consider the problem of determining the 


effectiveness of the antisubmarine device known as 
Squid. This is a device which throws three proximity- 
fuzed depth charges ahead of the launching ship in a 
triangular pattern. In order to simplify the problem 
we shall make the assumption that the heading of 
the submarine is known, and also the assumption 



Figure 3. Probability of sinking ship with spread of 
three torpedoes. 


that the aiming errors are distributed in a circular 
normal fashion, with the same standard deviation 
for all depths. We shall also assume that, if a single 
depth charge passes within a lethal radius R of the 
submarine, the submarine will be sunk. We wish to 
determine the best pattern for the depth charges. 



STANDARD DISPERSION <T 

Figure 4. Optimum spread as a function of dispersion of 
aiming errors. 

For any given pattern, the pattern damage func- 
tion depends on two variables, x and y , the aiming 
errors along and perpendicular to the course of the 
submarine. For any pair of values of x and y, 
Dp(x, y) is 1 if the submarine is sunk, and 0 other- 
wise. A typical case is shown in Figure 5. The origin 
is the point of aim, and the positions of the depth 
charges in the pattern are indicated by crosses. Each 
possible position of the center of the submarine is 
represented by a point in this plane. (Note that x and 
y are actually the negatives of the aiming errors.) The 


CONFIDENTIAL 


PATTERN FIRING-BALLISTIC DISPERSION PRESENT 


117 


three shaded regions represent the positions at which 
the submarine is destroyed by each of the three depth 
charges. The pattern damage function is 1 inside the 
shaded regions, and 0 in the unshaded regions. 


x 



Let f(x, y) dxdy be the probability that the center 
of the submarine be in the area element dxdy. Then 
the probability of destroying the submarine is 


P =J" D (x,y)f (x,y) dxdy 
- J^d f(%,y)dxdy . 


(16) 


In the last equation the region D of integration is 
just the shaded area in Figure 5. Because of the ir- 
regular shape of this region, analytical evaluation of 
this integral is impractical, and graphical methods 
must be used. In problems of this type a very con- 
venient aid is a form of graph paper known as “cir- 
cular probability paper.” This paper is divided into 
cells in such a way that, if a point is chosen from a 
circular normal distribution, the point is equally 
likely to fall in any of the cells. If an area is drawn on 
the paper, the chance of a point falling inside the area 
is proportional to the number of cells in the area. It 


follows that the integral of equation (16) can be 
easily evaluated by drawing the damage function to 
the proper scale on circular probability paper, and 
counting the cells in the shaded area. 

This method gives a rapid means of finding the 
probability of destroying the submarine with any 
given pattern. To find the best pattern, we note that 
changing the position of any one of the depth charges 
amounts to shifting the corresponding shaded area in 
Figure 5 parallel to itself to the new position. If three 
templates are made by cutting the outline of the 
shaded area for a single depth charge out of a sheet 
of transparent material, to the correct scale to go 
with the circular probability paper, then the best 
pattern can be found by moving the templates 
around on a sheet of circular probability paper until 
the number of cells within the lethal area is a maxi- 
mum. 


6.3 PATTERN FIRING-BALLISTIC 
DISPERSION PRESENT 

Up to this point we have neglected the fact that 
bombs, shells, and torpedoes do not hit the exact 
point they are aimed at, even if the aim is perfect. 
The dispersion of the errors arising from this source 
is called the ballistic dispersion of the projectiles. We 
shall always assume that these errors are indepen- 
dent of each other, and of the aiming error of the 
pattern as a whole. 

As a result of the ballistic dispersion we cannot tell 
the exact number of hits which will be obtained with 
a given pattern, even if we know the exact aiming 
error made. We can, however, calculate the proba- 
bilities of 0, 1, 2, • • • hits for each possible aiming 
error, and combining these with the damage coeffi- 
cients we can find the probability of destroying the 
target as a function of the aiming error. We thus cal- 
culate a pattern damage function which can be used 
exactly like the one we have treated previously for 
the case of no ballistic dispersion. 

To illustrate the effect of ballistic dispersion, let us 
reconsider the problem of the airplane dropping 
bombs on a railroad. Let us suppose again that two 
bombs are dropped, and that a hit by either within 
25 feet of the track will destroy the track. Let us sup- 
pose that the standard deviation of the error of the 
aiming of the stick is 100 feet and that there is a bal- 
listic dispersion of 25 feet for the bombs. 

Before calculating the damage function for a stick 
of bombs, let us first find the probability of destroy- 


CONFIDENTIAL 


118 


GUNNERY AND BOMBARDMENT PROBLEMS 


ing the railroad with a bomb aimed so that if there 
were no ballistic dispersion it would hit at a distance 
y from the center of the track. This is easily seen to be 

F(y) = — = / e~ xW25)2 dx 

V27r(25) J ~ y - 25 

(17) 

1 r-(v/ 25) +1 

= 4= / e ~ u2/2 du. 

V2t r-/ — (2//25) — l 

The form of this function is shown as the solid curve 
in Figure 6. [If there had been no ballistic error, F(y) 



Figure 6. Probability of destroying railroad with one 
bomb as function of aiming error (ballistic dispersion in- 
cluded). 


would have been unity for | y | < 25 and zero for \y\ 
> 25, as shown by the dotted curve in the figure.] 
Now if the bombs are dropped with stick spacing 
2a the values of y for the two bombs are x — a and 
x + a, where x is the bombing error. Since the ballis- 
tic dispersion of each bomb is independent of the 
other, the damage function is therefore 

D p (x) = 1 - {[1 -F(x-a)][ 1 - F(x + a)]}, 
D p (x ) = F(x — a) + F (x + a) — F{x — a)F{x + a ) . 

(18) 


The damage function for a = 25 feet is plotted in 
Figure 7 to illustrate its form. (If there had been no 
ballistic error, D v would have been unity for | x | < 50 
feet and zero for | x | >50 feet as shown in the figure.) 
When the damage function has been found, we take 
into account the distribution of the aiming errors, so 
that the probability of destruction is, by equation 
( 10 ), 

P(d)=-=J f D p (x)e~ x,/2(100> ’ dx . (19) 

V 2jr(100) J 

The evaluation of this integral is best made graph- 
ically. A plot of P{a) as a function of a then shows 


the best value of a, and the probability of success. 
Such a plot is shown in Figure 8, with the correspond- 
ing curve for no ballistic dispersion. It will be seen 
that the ballistic dispersion requires an increase in 



Figure 7. Damage function for stick of two bombs (bal- 
listic dispersion included). 


stick spacing from 50 feet to 85 feet, and has the 
effect of lowering the chances of success from 0.38 
to 0.34. 


6.3.1 Approximate Solution for Large Patterns 

The method of solution given in the previous sec- 
tion becomes very laborious for large patterns, par- 
ticularly if one has no previous idea as to the correct 



Figure 8. Probability of destroying railroad with stick 
of two bombs as a function of stick spacing. 


pattern to use. In this section we shall describe an 
approximate treatment of the problem which can be 
used to obtain solutions more quickly for large pat- 
terns. If more accurate results are required, this so- 


CONFIDENTIAL 


PATTERN FIRING-BALLISTIC DISPERSION PRESENT 


119 


lution may serve as a starting point for a solution by 
exact methods. 

Consider a coordinate system fixed at the center of 
the pattern. Because of ballistic dispersion there is 
probability f(x,y)dxdy that a projectile will hit the 
area element dxdy. The integral 

, y) dxdy (20) 


P can be increased by a positive choice of 8. If 

yi ) < y 2 ) , 

then P can be increased by a negative choice of 8. 
Hence for the function /(x, y) which makes P a maxi- 
mum, we must have 


is equal to the number of projectiles in the pattern. 
By changing the pattern we can cause extensive 
changes in the function /(x, y). Our approximation 
will consist of the assumption that /(x, y) can be 
changed arbitrarily by shifting the pattern, subject 
only to the condition that equation (20) remains 
satisfied. We can call / the pattern density Junction. 

If the lethal area of the target is L, then the ex- 
pected number of lethal hits on the target, if the cen- 
ter of the target is at x, y, is L/(x, y). Assuming that 
the number of lethal hits is given by the Poisson law, 
the probability of at least one lethal hit, and hence 
the probability of destroying the target, is 

D{x,y) = 1 - e - LKx ’ y) . (21) 

We shall therefore take this as our approximation to 
the pattern damage function. 

The total probability of destroying the target is 
then by equation (10) 

P = ff [1 -e- L ' M }f„(x,y)dxdy (22) 

where / p is the probability density for aiming the pat- 
tern, usually the normal density. We wish to find the 
function f(x, y) which maximizes P, subject to the 
condition (20). To determine the maximum let us 
consider the effect of increasing /(x, y) by a small 
amount 8 in an element at xi, yi, and decreasing 
(x, y) by an equal amount 8 at the point x 2 , y 2 . This 
obviously does not change the value of N in (20), and 
changes P by an amount (to the first order) : 


for all pairs of points at which /(x, y) > 0. Hence for 
all such points 


e- L ' M f p (x,y) = 'l>o, (23) 

where <£ 0 is some positive constant. 

Now let xi, 2/1 be a point, as before, where /(x, y) > 
0, and let x 3 , y 3 be a point at which /(x, y) = 0. Then 
if we decrease p(x i, 2 / 1 ) by a small amount 8 (which 
must now be positive) and increase f(x 3 , 2 / 3 ) by this 
amount; then P is increased by the amount 

[fp(x>, y 3 )-e~ LAx '’ vl) f p (xi, yi) ] Sdxdy 

= lfp(%h 2 / 3 ) — <t>o] Sdxdy . 


This is positive if f p (x 3 , 1/3) > <!>«■ Hence p(x, y) can- 
not vanish unless 


f p (x, y) < <t> 0 . 

From (23) and (24) we find our solution: 


fix, y ) 


jbn > fr( x ’ y i > ^ 


(24) 


(25) 


0, f p (x, y) < <#>o • 


The unknown constant <£o remains to be determined. 
It must be chosen so that (20) is satisfied, i.e., so that 


N-l 

L JJ L 4>o J 

f><t> 0 


dxdy . 


(26) 


{e- LKxiM f P {x h yf) - yi ) ] Sdxdy . 

When neither /(xi, 2 / 1 ) nor/(x 2 , 2 / 2 ) isO, [fix, y) cannot 
be negative, but may vanish] then, if 

e -Lf^ fp(Xh yi) > e - LtM f p {x 2 , yi) , 


In some cases 0 O can be found analytically; in others 
it must be found graphically by plotting the integral 
on the right of (26) as a function of <£ 0 . Compare this 
detailed analysis with the preliminary discussion in 
Section 5.2. 

The most important special case of this treatment 


CONFIDENTIAL 


120 


GUNNERY AND BOMBARDMENT PROBLEMS 


is that in which the probability density for aiming 
the pattern, f p (x, y), is the normal distribution : 

For this case (25) becomes 

/(*, „)-i [ -h( w.) 4($+ ~) - in *] - 


over the region where this is positive, and /(a:, y) = 0 
everywhere else. Using (26) to evaluate 0 O gives the 
final result: 


fix, y) = - 

Ju 


I LN + 

2'W Cy^/ . 


(27) 


over the region where this expression is positive, and 
/Or, 2 /) = 0 elsewhere. The mean density of the pat- 
tern, therefore, should be concentrated near the 
maximum of the aiming probability density, and 
should decrease parabolically as one goes out from the 
point of aim in any direction. 

In the one-dimensional case the corresponding so- 
lution is 


m 


1 r / 3 LN V/3 _ 1 
L L\4 V 2 J 2 <r 2 J * 


(28) 


Example of the Approximate Method 


As an illustration of the method of the previous 
section, let us consider the design of a depth charge 
pattern. Let us suppose that the errors of the attack 
are circularly normal, with a x = a y = 300 feet, and 
that proximity-fuzed depth charges are dropped or 
thrown. The lethal area of the submarine is 10,000 
square feet, and 13 depth charges are to be used. 

Putting the given constants into equation (27) 
gives 


10 4 / = 0.678 - 


18 X 10 4 ’ 


where r 2 = x 2 + y 2 . The value of / falls to 0 at r = 350 
feet, so the entire pattern should be enclosed within a 
circle of 350 feet radius. The number of charges 
which should be dropped within a radius r of center is 


N t 


-f; 


2i rrfdr 


= 2.13— - 0.0873 — . 


10 4 


10 8 


This function is plotted in Figure 9. To approximate 
this curve we may use the function indicated by the 
dotted line, which arises if we take a pattern consist- 
ing of one depth charge at the center, a ring of 6 



Figure 9. Number of depth charges inside radius r for 
13-depth charge pattern. 


depth charges at a radius of 150 feet, and a second 
ring of 6 depth charges at a radius of 250 feet. If 
these rings are staggered in a reasonable way we 
arrive at the pattern shown in Figure 10. (It should 


.1 

® o ® 



Figure 10. Possible depth charge pattern for maximum 
lethality. Standard error 300 feet. 


be noted that practical requirements on the number 
of throwers on destroyers may force a modification 
of this pattern. If the course of the destroyer is up 
the center of Figure 10, the two charges thrown 250 


CONFIDENTIAL 


PATTERN FIRING-BALLISTIC DISPERSION PRESENT 


121 


feet to the side would be replaced in actual practice 
by two more stern-dropped charges.) 

6.3.3 Probability Estimates by the Approximate 
Method 

If equation (25) is substituted into equation (22) 
we find, for the probability of destruction, 

P = [f(x, y) - <f> 0 ]dxdy , (29) 

where the integration is over the region in which 
/ > <t> o. If we put equation (25) into equation (20), w~e 
may write : 

NL f(x, y) - In </>o ] dxdy , (30) 


so that the probability of destruction is proportional 
to the number of projectiles. Figure 11 shows the 
exact relationship between P, (x 0 /a), and (NL/o). 



0.07 0.1 0.2 0.3 0.5 I 2 3 4 5 10 20 30 40 

NL 

a 

Figure 1 1 . Probability of destruction and size of pattern 
as function of number of projectiles — one dimension. 


(where the same region of integration is used) , we have 
two equations between which </> 0 can be eliminated 
to produce an equation connecting the probability 
of destruction, P, with the number of projectiles, N. 
This relationship is of great importance for making 
rapid estimates of P, or of the number of projectiles 
needed to obtain a desired value of P. 

The special case in which the aiming error has a 
normal probability density is so common that we will 


In the two-dimensional case, with 
1 


/ = 


2 7 T ( 7 xO” i 


exp 




the pattern extends over the region in which 
\<T X OyJ 



If ( xq / o ) is small, P is given approximately by 

= ~7= — > (33) 

3 V 7T \(T / V 27 T & 


Figure 12. Probability of destruction and size of pat- 
tern as function of number of projectiles — two dimen- 
sions. 


and the equations for P and N are 


P - 1 - (i + ir 0 ! ) 


,-(roV2) 


NL t r 4 
= 7 r ° • 

a x (Ty 4 


(34) 

(35) 


CONFIDENTIAL 


122 


GUNNERY AND BOMBARDMENT PROBLEMS 


When r 0 is small, P is given approximately by 


P — 

8 


NL 

2tT<J x d y 


(36) 


so that the probability of destruction is again propor- 
tional to the number of projectiles. The exact rela- 
tionship between P, r 0 , and (NL/a x a y ) is shown in 
Figure 12. 

As an example of the use of these curves, let us ob- 
tain an approximate probability of destruction for 
the depth charge pattern shown in Figure 10. Here 
N = 13, L = 10,000 square feet and <r x = a y = 300 
feet. Hence ( NL/a x <T y ) = 1.44. For this value, Figure 
12 gives r 0 = 1.2 and P = 0.14. The value r 0 = 1.2, of 
course, corresponds to the 350-feet pattern radius 
found in Figure 9. 


6.4 THE sampling method 

While the method discussed in the latter part of 
the foregoing section gives a rapid approximate solu- 
tion to pattern problems, its approximate nature 
makes it unsuitable for work in which precision is 
required. Since exact analytical solutions are usually 
impractical or very tedious to carry out, we present 
here a method which can, in principle, give any de- 
sired degree of accuracy, and which at the same time 
is of such a character that rough approximations can 
be found with little more effort than the approximate 
method just given. 

The method operates by a simple process of sam- 
pling. A pattern is selected for trial, and its proba- 
bility of destruction is obtained by repeated trials (on 
paper) of the pattern, observing on each trial whether 
the target is or is not destroyed. Each trial is con- 
ducted by selecting at random an aiming error from a 
suitably constructed artificial population of aiming 
errors. In addition, a ballistic error for each projec- 
tile is selected in a similar way from a population of 
ballistic errors. The position at which each projectile 
lands is thus found. From these positions the damage 
to the target is found. By taking a sufficiently large 
number of trials and averaging the results, the ex- 
pected damage for the pattern is found. By repeating 
this process for various patterns, the best pattern can 
be calculated. In the following sections we consider 
the technique of carrying out these processes. 


like, we shall use a table of random digits. Such a table 
may be constructed by any process which selects one 
of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 in such a way 
that each of the ten digits is equally likely to be se- 
lected, and the selection is in no way affected by the 
result of previous selections. One way of selection 
might be, for example, to mark ten identical balls 
with the ten digits, and form the table by drawing the 
balls from an urn, replacing the ball after each drawing 
and mixing well between drawings. Another method 
would be to roll a die twice for each digit, translating 
the rolls into digits according to the following table: 

Second Roll 



1 

2 

3 

4 

5 

6 

1 

1 

2 

3 

4 

5 

6 

2 

7 

8 

9 

0 

1 

2 

3 

3 

4 

5 

6 

7 

8 

4 

9 

0 

1 

2 

3 

4 

5 

5 

6 

7 

8 

9 

0 

6 

X 

X 

X 

X 

X 

X 


If the first roll is a 6, the die is rolled again. It should 
be obvious that if the die is true, the ten digits are 
equally likely to be chosen. (The reader is warned 
that if two dice are thrown together, it must be pos- 
sible to distinguish the two, marking one No. 1 and 
the other No. 2.) 

It is most convenient to use one of the published 
tables, such as Tippett’s, 14 which have been very 
carefully examined for randomness. A short table 
of random digits is given at the back of the book as 
Table I. 

If we now wish to sample a stochastic variable x 
(for example a ballistic error), we may take a se- 
quence of random digits, preceded by a decimal point, 
as a value of the distribution function F (as defined 
in Chapter 2) for the variable. The functional rela- 
tionship between x and F is therefore a method of 
converting from a table of random digits to a sample 
population of the stochastic variable x. 

To illustrate, let us sample a normal distribution. 
The distribution function is 


6 4.1 Construction of Sampling Populations 

As a basis for the construction of our sampling 
populations of aiming errors, ballistic errors, and the 



v 27 r 


If we take the first five groups of five digits in Table I 


CONFIDENTIAL 


THE SAMPLING METHOD 


123 


as values of F, we obtain the following results, by 
using Table V to convert from F n to x. 


F 

(x/a) 

0.57705 

0.19 

0.13094 

-1.13 

0.60835 

0.36 

0.36014 

-0.36 

0.35950 

-0.36 


Continuing in this way a table of random values of 
(x/a) can be constructed. Two such tables are given 
at the back of the book. Table II is a table of random 
angles (in degrees), while Table III is a table of 
random normal deviates. 


6.4.2 A Rocket Problem 

As a first illustration of the sampling method, let 
us consider the following problem: A plane fires two 
rockets at a circular gun emplacement from a 30- 
degree glide at a range of 1,500 yards. The gun em- 
placement is R feet in radius, and is destroyed if 
either rocket hits it. The rockets are fired from par- 
allel rails, 10 feet on either side of the center line of 
the plane. The aiming error is normal, with a x = a y = 
10 mils, and the ballistic error of the rockets is also 
normal, with a x = a y = 5 mils. We wish to compute 
the probability of destroying the target. 

First translating the mil errors into lengths: at 
1,500 yards a 10 mil error is 15 yards or 45 feet, while 
a 5 mil error is 22.5 feet. These are the standard 
errors in a plane perpendicular to the line of flight of 
the rockets. On the ground, the range errors are in- 
creased by a factor of cosec 30 degrees = 2. This in- 
creases the aiming range error to 90 feet, and the 
ballistic range error to 45 feet. We thus have the fol- 
lowing standard deviations : 

Aiming Range a xa = 90 feet 

Deflection a ya = 45 feet 

Ballistic Range a xb = 45 feet 

Deflection a yb = 22.5 feet 


If xi, 2/1 are the coordinates of the landing point of the 
starboard rocket, then 


Xi = Xa + Xbl , 

2/1 = 2/a + 2/61 + 10 , 


where x a , y a are the aiming errors of the salvo, x bh y b i 
are the ballistic errors of the rocket, and the 10 is the 


displacement of the rails from the center line. The 
coordinates of the landing point of the port rocket 
are 

X>2 = X a “H X b 2 , /oq\ 

2/2 = 2/a + 2/62 - 10 . ^ ; 


Each salvo therefore requires the sampling of six 
numbers : x a , y a , x b i, 2/6i, x b2 , y b 2 from normal popula- 
tions with the standard deviations given above. 


zzo 

1 1 1 1 1 1 1 

— 1 — 1 — 1 — 1 — ®l — 1 — 1 — 

200 


- 

180 


- 

160 


- 


o o 


140 

X 

O X 

<r 120 


_ X 

HI 


© 

o IOO 


X 



X 

80 


o 

60 

o 

X 

40 

O i 

© 

20 


X 


X X 

X X 

0 

A 



X 


20 


X 



o 

40 

X 

o ° 


o 


60 

o 

o 


o 

X 

80 


- 

£ IOO 
§ 

o 

X 

© 

X 

to 120 

- 


140 

- 

- 

160 

_ 

- 



© 

180 

- 

- 

200 

_ 

- 

pt >n 

o 

1 1 1 1 1 1 1 

1 1 1 1 1 1 1 


160 120 80 40 0 40 80 120 160 


LEFT PIGHT 

Figure 13. Sample pattern of salvos of two rockets. 

Nearer rocket of each pair is marked with cross, farther 

with circle. 

The work of sampling is shown in the computation 
sheet in Table 1. A sample of 20 salvos is illustrated 
(rather smaller than should be used in practice) . The 
first column ( x a ) is obtained by taking the first 20 
normal deviates in Table III, and multiplying each 
by the standard deviation, 90 feet. The next five 
columns are obtained from the following sets of 20 
normal deviates, multiplying them by the appro- 
priate cr’s. The next four columns are obtained by 
using equations (37) and (38) . 

In Figure 13 the positions of the rocket hits are 
plotted. In each salvo, the rocket hitting closer to 
the center point is marked with a cross, the farther 
being marked with a circle. The distance of the closer 
rocket from the center is entered in the eleventh 


CONFIDENTIAL 


124 


GUNNERY AND BOMBARDMENT PROBLEMS 


Table 1. Calculation of probability of hit by sampling. 


X a 

y« 

Xbl 

Vbi 

xm 

Vb2 

Xi 

Vi 

x 2 

2/2 

R 

Rank 

72 

7 

-67 

-20 

-40 

-15 

5 

-3 

32 

-18 

6 

1 

-62 

-27 

21 

-30 

6 

-45 

-41 

-47 

-56 

-82 

62 

11 

34 

33 

-58 

3 

40 

-12 

-24 

46 

74 

11 

52 

9 

12 

52 

-61 

-25 

-6 

15 

-49 

37 

6 

57 

58 

10 

156 

17 

-37 

-3 

-18 

1 

119 

24 

138 

8 

122 

16 

-49 

88 

-22 

-5 

-48 

14 

-71 

93 

-97 

92 

117 

15 

-19 

71 

-12 

26 

24 

-16 

-31 

107 

5 

45 

45 

8 

-54 

-54 

47 

5 

-21 

23 

-7 

-39 

-75 

-41 

40 

7 

-143 

-66 

-64 

-5 

36 

-45 

-207 

-61 

-107 

-121 

162 

19 

-54 

16 

17 

28 

52 

-21 

-37 

54 

-2 

-15 

15 

2 

38 

84 

-131 

5 

21 

26 

-93 

99 

59 

100 

116 

14 

150 

63 

69 

4 

-11 

47 

219 

77 

139 

100 

172 

20 

60 

-17 

-23 

6 

0 

14 

37 

-1 

60 

-13 

37 

6 

5 

-11 

-46 

17 

27 

18 

-41 

16 

32 

-3 

32 

5 

123 

-11 

-35 

22 

25 

-17 

88 

21 

148 

-38 

90 

12 

-43 

28 

77 

8 

66 

-4 

34 

46 

23 

14 

27 

3 

-102 

53 

-4 

1 

-69 

-21 

-106 

64 

-171 

22 

124 

17 

45 

11 

58 

-16 

68 

13 

103 

5 

113 

14 

103 

13 

-17 

-11 

-43 

10 

24 

-7 

-60 

9 

7 

-28 

29 

4 

106 

-14 

41 

-51 

32 

-5 

147 

-55 

138 

-29 

141 

18 


column of the computation sheet (headed R). Finally 
the salvos are ranked in order, the salvo with the 
closest hit being numbered 1, the next 2, and so on. 
It is now a simple matter to plot the number of 
salvos with at least one hit inside a target of radius 
R, as a function of R. The result is the step curve 
shown in Figure 14. Finally the step curve is smoothed 
as shown in the same figure. 

To estimate the precision of this curve, we may 
calculate the standard deviation of the value of P, 
the probability of at least one hit. If the true value 
for any value of R is P 0 , the standard deviation of the 
value for a sample of n is 


Fo(l - Po) . 

n 

Since P 0 (l — Po) ^ 1/4, the standard deviation is 
everywhere less than 1/2 y/n. For our sample of 20 
this has the value 0.1 1. The step curve may therefore 
be expected to depart from the true curve by amounts 
of the order of 0.11, measured along the P axis. It 
will be noticed that the smooth curve departs from 
the step curve in Figure 14 by amounts of this order. 



TARGET RADIUS IN FEET 


Figure 14. Probability of at least one hit as function of 
target size, from sample shots of Figure 13. 

The smoothing undoubtedly reduces the sampling 
error, but it is not possible to estimate how much. 
Obviously the precision can be increased to any de- 
sired extent by taking a large enough sample, 
although the precision increases only slowly with n. 

6.4.3 Short Cuts in the Sampling Method 

There are many short cuts and tricks which may be 
used to lighten the work of the sampling method. 
These usually depend on special features of the prob- 


CONFIDENTIAL 


THE SAMPLING METHOD 


125 


lem to be solved, so that no general rules can be 
given. It may, for example, be possible to determine 
the damage function of a pattern analytically, but 
not to carry out the final integration. In such a case 
the final integration can be made by a sampling 
process. In other cases there may be independent 
intermediate probabilities which can be found by 
sampling methods, and the results combined by 
analytical methods. The recognition and use of such 
devices depends on the skill and ingenuity of the 
worker, rather than on previous knowledge. 

A very common device is illustrated by the follow- 
ing example : Suppose that an exact evaluation of the 
depth charge pattern of Figure 10 is needed. To 
carry out the calculation by direct sampling requires 
the selection of 29 sample numbers for each trial: two 
for each depth charge to determine its ballistic error, 
two for the aiming error, and one to determine the 
orientation of the submarine. The work, however, 
may be shortened in the following way. A master 
chart of the depth charge pattern is prepared. Ten 
sample ballistic errors for each depth charge are then 
found, and the resulting actual positions of the depth 
charges are marked on the chart, using different 
colors or symbols to designate the position the depth 
charge was supposed to have. For instance, around 
the point A in Figure 10 (the point where the charge 
would fall if there were no ballistic error) will be a 
scatter of ten points, which can be labeled A 1, A2, 
• • •, A 10; and around point 5 is another scatter, 
labeled 51, 52, • • • ,510. 

When this drawing is completed, samples of posi- 
tions and orientations of the submarine are drawn. 
This may be rather complicated if one wishes to take 
into account the maneuverability of the submarine 
when computing its distribution function (for in- 
stance in some cases of interest one knows the posi- 
tion and velocity of the submarine a certain time 
before the pattern is dropped, but does not know its 
maneuvers thereafter) : in any case a random sample 
of possible positions and orientations is drawn. By 
means of a template showing the outline of the lethal 
area of the submarine (or, if one is finicking, the 
contours of equal probability of destruction D ) each 
sample submarine position is examined for hits, and 
the probability of destruction can be found for that 
position. 

For example, if the template in one of the sample 
positions shows that 2 of the 10 points around point 
A (A3 and A 6, for instance) and 1 of the 10 points 
around 5 (51, for instance) are inside the lethal area, 


and no others, then the probability of destruction in 
that position is recorded as 

‘-(■-roM'-fo)- 028 - 

If 100 sample positions are taken, a good approxi- 
mation to the desired probability will be found by 
averaging the probabilities in the various positions. 
The whole process involves 560 sampling numbers, 
whereas 100 direct trials would take 2,900 sampling 
numbers. Needless to say, auxiliary tables for the 
combination of probabilities should be constructed 
and used. 

6 . 4.4 Train Bombing 

Another example of the use of the sampling method 
is in the calculation of the probability that a stick 
of bombs will damage a target. As an example, let us 
take the case where the ballistic deviation (in range 
and deflection) equals the aiming deviation in deflec- 
tion, and equals one-half the aiming deviation in 
range. This is a somewhat greater ballistic error than 
occurs in practice, but it has been taken large to 
contrast with our earlier calculations, where we 
assumed zero ballistic deviation. 

Using Table III, we plot the mid-points of 25 sal- 
vos, using a range scale twice as large as the deflec- 
tion scale. Then, with further use of the table, we 
displace four points from each mid-point, thus arriv- 
ing at a plot of 100 points, 25 salvos of 4 bombs 
each. This is shown in Figure 15. The individual hits 
are represented by the numbered circles; the indi- 
vidual salvos have their 4 circles connected by 
straight lines and are labeled by letters a to y. Here 
the ballistic error is large enough so that the patterns 
overlap. 

To compute the probability of hitting a target of 
known size and shape, when bombed by four bombs 
in salvo, we superpose on this pattern a drawing of 
the target having a size corresponding to the ratio 
between the actual target size and the normal devia- 
tions, an orientation related to the direction of the 
plane’s track over the target and having the point of 
aim at the center of the figure. 

A mere counting of the circles whose centers are 
inside the target gives the number of bombs hitting 
the target in 25 passes. Dividing by 25 gives the 
chance of a bomb hitting in one pass. 

Some of the hits may be two (or three or four) out 


CONFIDENTIAL 


126 


GUNNERY AND BOMBARDMENT PROBLEMS 



Figure 15. Random salvos a to y of 4 bombs each. Ballistic dispersion equals deflection dispersion equals one-half 
range dispersion, R. 1 H 


CONFIDENTIAL 


THE SAMPLING METHOD 


127 


Table 2. Bombing calculation by sampling method ( D = 0.5). 


Stick spacing 

Zero 

W 2) 

R 

Ni, no. sticks with only 1 hit in an area. 

Does not apply 

17 

21 

Nu, no. sticks with 1 hit in each of 2 areas. 

U 

3 

1 

Nm f no. sticks with 1 in each of 3 areas. 

it 

1 

1 

N 21 , no. sticks with 1 in one area, 2 in another. 

(( 

3 

0 

N 31, no. sticks with 1 in one area, 3 in another. 

u 

1 

0 

No. salvos which result in 1 hit ni = 6A1 + 8 Nu + 9iVm + 10iV 2 i + 12iV 3 i 

4 

177 

143 

No. salvos which result in 2 hits n 2 = 2N n + 3 Nm + 4iV 2 i + 6iV 3 i 

6 

27 

5 

No. salvos which result in 3 hits n 3 = Nm 

0 

1 

1 

Total no. sticks in sample = N 

25 

600 

600 

Expected total hits per salvo (or stick) = (l/iV)(ni + 2 n 2 + 3n 3 ) 

Expected fraction salvos which resulted in at least 1 hit 

0.640 

0.390 

0.260 

= (1 /N) ( ni + n 2 + n 3 ) 

0.400 

0.342 

0.248 

Probability of sinking ship = (1 /N) (0.5ni + 0.75n 2 + 0.875n 3 ) 

0.260 

0.183 

0.127 


of one salvo. Such pairs (or triplets or quadruplets) 
should only count as a single, if we wish to compute 
the chance that the target is hit at least once in a 
salvo. On the other hand, if the target is a ship, with 
probability of sinking given by equation (2), the 
chance of sinking the ship with a single salvo is ap- 
proximately 

25 

+ n 3 [ 1 — (1 — D 3 ) ] + n 4 [ 1 - (1 — D) 4 ] }, 

where Ui is the number of salvos having only one hit 
inside the target, n 2 the number having two hits 
inside, etc. When a target of the shape and size 
shown in Figure 15 is placed over the point of aim, 
the count is that shown in the column marked Zero 
(stick spacing) in Table 2. The other two columns will 
be explained below. 

To use the same figure for a stick of four bombs, 
we can imagine displacing each circle marked 1 by 
1.5 times the stick spacing downward, the circles 
marked 3 by 0.5 times the spacing upward and those 
marked 4 by 1.5 times the spacing upward. Rather 
than redraw the figure, we can draw 4 targets, as 
shown by (A, B, C, D), displaced in the opposite di- 
rection by similar amounts; and count all the circles 
marked 1 which are in area A, all circles marked 2 in 
area B, and so on. 

Of course we could have considered the circles 
marked 2 as being the first bombs in the train; in 
which case we would have counted the number of 
circles marked 2 in area A, etc. 


There are 24 different ways that (1, 2, 3, 4) can fit 
into areas (A, B, C, D); so that we can consider the 
samples as representing 24 X 25 = 600 trials. These 
trials are not all independent of each other, so that 
the result will not be as accurate (on the average) as 
if we had drawn 600 independent salvos; but the 
result will be less subject to fluctuations than if we 
had taken only the number of l’s in A, 2’s in B, etc. 

The rules for computing the result are worked out 
by setting down all 24 permutations and counting 
which of the permutations corresponds to what re- 
sult. They are: 

1. We count the number of circles in each area 
which belong to sticks having hits in only one area 
(for instance stick e has e4 in area A and no other 
in another area; and stick u has u2 and w4 in area C 
but none in another area). Call this number Ni. 

2. We count the number of pairs of circles belong- 
ing to one stick, one of which is in one area and one 
in another (for instance stick w has w2 in area C and 
w4 in D ) . Call this number Nu. 

3. Count the number of triplets belonging to one 
stick, two of which are in one area and one in another 
area. (If wS were displaced slightly upward, it also 
would be in area D, along with w4.) Call this number 
Nu. 

4. Count the number of triplets belonging to one 
stick, each of which is in a different area (d2 is in 
area B, d4: in area C, and d3 in area D). Call this 
number Nm. 

5. The extension of these definitions to N 22 , Nun, 
etc., should by now be obvious. 

We then count over the various 24 permutations of 


CONFIDENTIAL 


128 


GUNNERY AND BOMBARDMENT PROBLEMS 


1, 2, 3, 4 which make the above combinations into 
single, double, etc., hits, and arrive at the following 
rules : Out of the 600 sample sticks : 

The number of sticks resulting in a single hit is 

U\ = 6iVi + 8 An T" 9A m T IOJV21 

+ 9 Aim + IOA2II + 12^31 + 8A^22- 

The number of sticks resulting in two hits is 

n<i = 2Nu + 3 Am + 4A21 + 6Ami 

+ 4^211 + 6 A 31 + 8A^22- (39) 

The number of sticks resulting in three hits is 

713 = Am + 2A 2 m 

The number of sticks resulting in four hits is 

714 = Nun. 

In the case shown in Figure 15 the circles inside 
the target areas are 

A; a3, c4, ml. 

B;d2,f3,f4,hl,j3,kl. 

C; 64, d4, ol, o2, q3, qi, r3, si, u2, w4, w2, x3. 

D ; d3, p4, A, v3, wk. 


The numbers N are therefore 

Ni = 21; A — a, e, nr, B — /( 2), h, i, k;C — b, 
o( 2), 9(2), r, s, u( 2), x;D — p, t, v. 

Nu = 1; C, D — w. 

N in = 1 ', B , C , D — d. 

These are entered in Table 2 in the third column and 
the resulting calculations are obvious. Another count 
was made for a spacing of 0 .5R, and the results 
shown. It can be seen that, for such a large ballistic 
dispersion compared to the aiming dispersion, it is 
somewhat better to drop the bombs in salvo rather 
than with any appreciable stick spacing. The ballistic 
dispersion does enough spreading without needing 
any additional amount. 

Similar calculations can be made for the target 
areas rotated about their centers, to determine the 
effect of approach bearing on the probabilities. When 
the areas overlap, circles in the common area must 
be counted in both areas. The same chart can be used 
for sticks of three or two, by revising equations (39) . 


CONFIDENTIAL 


Chapter 7 


OPERATIONAL EXPERIMENTS WITH EQUIPMENT 

AND TACTICS 


I n time of war the measures of effectiveness of 
weapons used in various ways must ultimately be 
determined from experience on the field of battle, for 
only in this way can we be sure of their actual behav- 
ior in the face of the enemy, and only then can we 
devise tactics which we can be sure are effective in 
practice. Constant scrutiny of operational data is 
necessary to see whether changes in training pro- 
cedures make possible more effective utilization, and 
to see whether changes in enemy tactics require 
modifications of ours. 

However, operational data are observational, rather 
than experimental data. Conditions cannot be 
changed at will, pertinent variables cannot be held 
constant, and the results give overall effectiveness 
with usually little chance to gain insight into the 
adequacy of various components of the tactic. The 
check, by operational data alone, of an analytical 
theory of the effectiveness of a given tactic, is not 
often detailed enough to be able to determine the cor- 
rectness of all the component parts of the analysis. To 
obtain such a confirmation, an independent variation 
of each of the component variables is always desirable 
and usually necessary, a procedure which the enemy 
is seldom kind enough to allow us to carry out on the 
field of battle. 

To gain insight into the detailed workings of a 
given operation, therefore, so that one can redesign 
tactics in advance of changing conditions, it is neces- 
sary to supplement the operational data with data 
obtained from operational experiments, done under 
controlled conditions with a specially designated task 
force. These additional data can never take the place 
of the figures obtained from battle, for one can never 
be sure what the enemy is likely to do, or how our 
own forces will react to battle conditions. Never- 
theless, suitable experimental data, obtained under 
controlled conditions approximating as closely as 
possible to actual warfare, can be of immense assis- 
tance in providing more detailed knowledge of the 
complex interrelations between men and equipment 
which make up even the simplest operation. They 
are the only available data during peacetime, so it is 
important that they be gathered as carefully as 
possible. 


This idea of operational experiments, performed 
primarily not for training but for obtaining a quanti- 
tative insight into the operation itself, is a new one, 
and is capable of important results. Properly imple- 
mented, it should make it possible for the military 
forces of a country to keep abreast of new technical 
developments during peace, rather than to have to 
waste lives and energy catching up after the next war 
has begun. Such operational experiments are of no 
use whatever if they are dealt with as ordinary tac- 
tical exercises, however, and they must be planned 
and observed by trained scientists as valid scientific 
experiments. Here, then, is an important and useful 
role for operations research for the armed forces in 
peacetime. 

7.1 PLANNING THE OPERATIONAL 
EXPERIMENTS 

The carrying out of operational experiments is a 
difficult problem, because of the large number of 
variables involved, but the fundamental principles 
are the same as for any other scientific experiment. 
The results to be aimed at are a series of numerical 
answers, representing the dependence of the measures 
of effectiveness on the pertinent variables. The be- 
havior of these variables should be known, and they 
should be varied independently, as far as possible, 
during the experiment. Since operational behavior 
depends on the crew as well as the equipment, the 
state of training of this crew should be investigated. 
In fact, the learning curves for the crews should be 
determined as fully as possible, so that as one changes 
from crew to crew the effect of training can be taken 
into account. These measurements on training will 
also be valuable in indicating the amount of training 
which will be necessary when the equipment is put 
into actual operation. The maintenance problems of 
the equipment must also be investigated, and simple 
checks must be found to determine the state of main- 
tenance of the gear during each portion of the test. 

Since the tests are to determine the behavior of 
equipment and men, average crews must be used to 
handle all gear entering into the operation. The scien- 
tific observers must confine themselves to observing, 


CONFIDENTIAL 


129 


130 


OPERATIONAL EXPERIMENTS WITH EQUIPMENT AND TACTICS 


and should not interfere in the operation itself. If, for 
instance, an antiaircraft fire director is being tested, 
the usual crew must be put in charge of the director 
and of the gun, and the usual orders given them. This 
crew must not be allowed to know any more about 
the position of the incoming test plane than do usual 
crews in combat. The observers should occupy them- 
selves with photographing, or otherwise recording, 
the actions of the crew and the results of the firing. 
There should be many observers specially trained for 
this task, but the observers must keep themselves 
outside the experiment itself. 

7.1.1 Preliminary Theory 

Another important requirement is that one should 
have some general theory of the operation before the 
experiment is started. This requirement is in common 
with other scientific experiments; one does not 
usually blindly measure anything and everything 
concerned with a test, one usually knows enough 
about the phenomena to be able to say that such and 
such variables are the crucial ones, and that the effect 
of others is less important. One should know approx- 
imately where the errors are likely to be the largest, 
and should be able to get the range of the variables 
over which the greatest number of measurements must 
be made. It is not necessary that the theory be com- 
pletely correct, for the theory merely provides a 
framework for planning the experiments. If the meas- 
urements turn out to disagree with the theory, this 
will be almost as helpful as if they agreed. In fact, 
an investigation of the disagreement between the 
measurements and the preliminary theory sometimes 
provides the most fruitful results of the whole experi- 
ment. 

Not only should there be many extraoperational 
observers to keep track of all of the variables in- 
volved, but there also should be enough computers 
available so that the data can be reduced as fast as 
the experiment goes ahead. It is extremely dangerous 
to take all the data without reducing any of them and 
to allow the crews and equipment to disband before 
any results are obtained. It is practically certain that 
the results will show that certain further measure- 
ments should have been made, measurements which 
could have easily been made while the equipment was 
assembled, but which are extremely difficult to ob- 
tain later. A continuing, though preliminary, analysis 
of the results can tell when the measurements are in- 
adequate, when a new crew has not been sufficiently 


trained, etc., and can indicate deviations from the 
preliminary theory so that this can be analyzed in 
time. 

7.1.2 Preliminary Write-up 

In drawing up a plan for an operational experiment 
it has been found by experience that the following 
five items should be written down in detail : 

1. Subject. 

2. Authorization. 

3. Purpose and Aims of Testing. 

4. Present Status of Available Data. 

5. Plan of Procedure. 

Items 1, 2, and 3 are self-evident requirements. Since 
the work concerns equipment already in existence 
there will be certain performance data already ac- 
cumulated by the technical experts who were re- 
sponsible for its development and production. If the 
equipment is newly designed the chances are that the 
technical experts had been told some of the require- 
ments and tactical uses. Therefore, there exists some 
performance data from the development and accept- 
ance stage. If the equipment has already been in 
service, the evaluation called for would either be in 
connection with some new tactical use or due to un- 
favorable reports from the field. Thus there exist 
known facts about its performance, and these should 
be assembled under item 4 of the above outline. The 
Plan of Procedure called for in item 5 needs some 
explanation. 

7.1.3 Plan of Procedure 

This calls for an itemized account of how the 
needed data are to be obtained. It is a detailing of 
the duties of various observers, of their training be- 
forehand, etc. It has been found expedient in this 
type of work to gather more data during the tests 
than might seem necessary. The old adage of “penny 
wise and pound foolish” should be remembered and 
a really adequate program carried out. It is a cardinal 
principle that facts are recorded and not impressions, 
so every possible use is to be made of stop watches, 
range finders, thermometers, etc. The limitations of 
all equipment must be determined. This is not in 
order to find a fault, but to govern tactics. 

A generous use of movie cameras and still cameras 
has been found extremely valuable when used prop- 
erly, i.e., camera focal length, frame frequency, and 
altitude, must be known and recorded if data are to 


CONFIDENTIAL 


PLANNING THE OPERATIONAL EXPERIMENTS 


131 


be taken from camera evidence. Thirty seconds of 
movies showing equipment performance is more con- 
vincing to fleet commanders than volumes of reports. 
All these things and more must be thought through 
well in advance, so that the testing goes off smoothly. 

Experience has shown that the data must be 
gathered in such a way that they can be understood 
during the testing period, even if this requires a 
group of men doing nothing but recording, tabulat- 
ing, computing, and constructing graphs. In this 
way the technical director of the project knows 
whether his data are significant, and whether he has 
completed testing before the experimental facilities 
are diverted to other efforts. One other reason for 
this: should the data from a given set of conditions 
give an unexpected result, the experiment with these 
conditions can be repeated at once. Finally, in the 
plan for procedure some consideration for the time 
of completion and urgency of the project should play 
a role. Interim instructions or a report of first findings 
may expedite a fleet commander’s decision. 

7.1.4 Significant Data 

Keeping in mind the tactical use of the equipment 
should in itself insure that the data taken are sig- 
nificant. At least one person with field experience 
should be assigned to the project for this reason. A 
few examples will indicate how one must distinguish 
significant data from other kinds. 

In rocket firing from an aircraft with a newly de- 
veloped sight a pilot is assigned a plane, and proceeds 
to fire several hundred rounds on a land range over a 
period of several weeks. Do the data from this effort 
constitute a measure of equipment performance from 
an operational point of view? Obviously there are 
reasons to doubt that equipment performance would 
be the same as it would be under battle conditions. 
Operational data on such a sight should come from 
several squadrons with respectively different types 
of aircraft. The rockets should be fired on a towed 
target at sea to prevent range or azimuth correction 
made possible by fixed objects on a land range. In 
addition, the pilots should fire at the target from 
several directions relative to the wind, and the firing 
should be spread over enough days to allow for rea- 
sonable variations in wind and sea conditions. In 
other words, to evaluate the rocket-sight performance 
the project officer must keep in mind that the rocket 
sight will be used tactically in such a way that the 


aircraft will make one or, at most, two passes at the 
target. 

An evaluation of the experimental data should 
have some tactical conclusions. The fighting forces 
and the planners will find conclusions on the per- 
formance of the sight useful in order to judge how 
many planes are required to knock out a given target. 
It would also be the duty of the operations research 
worker to seek further confirmation of performance 
of the sight from combat information, if this is pos- 
sible. 

Sometimes it is not necessary to get such a spread 
of data. Let us consider the firing of torpedoes from a 
destroyer, and the evaluation of the performance of a 
new torpedo. The data needed are the comparative 
performance with other torpedoes under various tac- 
tical conditions, and the data can be obtained by one 
or two destroyers. Operational success can then be 
inferred, since data on the launching errors are al- 
ready available. A similar study might be adequate 
in the case of airborne torpedoes. 

Considerable care must be taken that the data 
from performance of sound or radar equipment are 
significant. In the case of sound gear, the performance 
is influenced by temperature gradients, density 
layers, and depths of the water. In the case of radar, 
the performance is altered by moisture, temperature 
gradients, etc. Thus care must be taken that data 
reported on performance are properly qualified, so 
that tactics designed for their use will be properly 
varied to meet the conditions. In addition to the 
tricks played by the elements on such gear, there is 
the large variable of equipment maintenance by fleet 
personnel. Every effort should be made to allow the 
equipment to age or even disintegrate under the 
maintenance of those personnel whom the fleet 
expects to service it. Having done this, the tests 
should then be conducted for performance. In this 
way it will be possible to assess the maintenance 
effort required by the fighting forces to keep the 
equipment effective, and thus to estimate whether 
the advantages of the new gear are sufficient to war- 
rant the maintenance and training effort. 

7.1.5 Conclusions 

The data gathered in evaluating equipment for 
tactical use will be useless to the operational com- 
mands unless there is, in the report, an interpretation 
of the data and conclusions. The theater commander, 


CONFIDENTIAL 


132 


OPERATIONAL EXPERIMENTS WITH EQUIPMENT AND TACTICS 


faced with peculiar conditions, may not agree with 
the conclusions in the report, but a summing up, 
along with the data, will help him reach the correct 
conclusions for his theater’s requirements. 

7.2 ACCURACY MEASUREMENTS 

One of the commonest types of test programs is 
that in which the accuracy of a weapon is to be 
measured. Such programs may be test firings of 
rockets to determine the distribution of aiming and 
ballistic errors, practice firing of guns to determine 
gun and director errors, practice runs by destroyers 
on submarines to determine the errors in dropping 
depth charges when various approach tactics and 
attack directors are used, test firings of torpedoes 
from submarines, and so on. In each case there is a 
target at which projectiles are fired, and in each case 
the measure of effectiveness desired is the distribu- 
tion of the projectiles around the target. In many 
cases it is also necessary to analyze the sources of 
error with the object of improving the accuracy. 

Whenever it can be arranged (it is not always pos- 
sible), arrangements should be made to record the 
position of each projectile fired relative to the target. 
This is not usually easy, and requires the closest co- 
operation with experimental laboratories and with 
operating personnel. Sometimes when a land target 
is used, the positions of the hits can be measured di- 
rectly by a surveying party after each salvo. When 
the target is on the surface of water, photographic 
recording is usually used. (When possible a Fleet 
Camera Party should be used.) The bursts of A A 
shells around air targets have also been recorded 
photographically. With underwater targets, under- 
water sound methods can sometimes be used. The 
interpretation of photographs and the calculation of 
underwater sound data both require careful study if 
accurate measurements are to be made. Even when 
direct measurements are made on land, it is fre- 
quently necessary to introduce corrections for un- 
even terrain. 

In order to separate the effects of aiming errors 
and ballistic errors, the projectiles should be fired in 
salvos of at least two at a time. Larger salvos would 
be better in some cases, but considerations of econ- 
omy frequently keep the salvo down to two projec- 
tiles. However, unless it is known that either the aim- 
ing error or the ballistic error can be neglected (very 
rarely the case), salvo firing is absolutely essential. 

In recording the data, the salvo to which each pro- 


jectile belongs must be recorded. This is most easily 
done by means of a salvo number. It should go with- 
out saying that all other pertinent data should be 
recorded : target, ship or plane firing, wind or weather 
conditions if relevant. 

7.2.1 Calculation of Standard Deviations 

The first step in working up the data is to calculate 
the mean point of impact [MPI] of each salvo, by 
averaging the positions of the projectiles in the salvo. 
The distribution of the projectile positions about the 
MPI is obviously independent of the aiming error, 



Figure 1. Use of probability paper. Deviations from 
MPI, <x = 18, <r 6 = 18V2 = 25. 


and so may be used to determine the ballistic error. 
Ballistic errors are almost always normally distri- 
buted (but not necessarily circularly), and it is easily 
shown that if the errors are normally distributed 
about the point of aim, they are also normally dis- 
tributed about the MPI, with a standard deviation 
which is less than t hat about the point of aim by a 
factor y/ (n — 1) /n, where n is the number of pro- 
jectiles in the salvo. 

The standard deviations of the ballistic errors in 
range and deflection are most easily found by the use 
of “probability paper.” This paper is ruled with a 
linear scale along one axis, and graduated on the 
other axis according to the normal distribution func- 
tion F n {x) (see Chapter 2). To use the paper, the 
fraction of a population of values of a stochastic 
variable £ with values less than a given value x, is 
plotted against x, plotting the value of x on the linear 
scale and the fraction on the F n scale. If the popula- 
tion has a normal distribution, the result will be a 
straight line. The mean value is then found at the 
point F n = 0.5000, while the standard deviation is 


CONFIDENTIAL 


ACCURACY MEASUREMENTS 


133 



the difference between this value and the value cor- 
responding to F n = 0.8413. 

A plot of this kind for the difference in range (or 
deflection) between each projectile and the MPI of 
its salvo, as shown in Figure 1 , therefore gives an 
immediate test of whether the errors follow the nor- 
mal distribution law, and, if the normal law is obeyed, 
gives the mean (which will, of course, be zero) and 
standard deviation. The standard deviation a b of the 
ballistic errors about the point of aim instead of the 
MPI is found by multiplying the MPI by \/n/ (n — 1 ) . 

If probability paper is not available, the standard 
deviation of the errors from the MPI may be found 
by the arithmetic method of finding the square root 
of the mean of the squares of the deviation. This 
process, however, is laborious, and does not check 
the normality of the population. 

When the distribution of ballistic errors has been 
determined, the distribution of aiming errors can be 
found from the distribution of the MPPs as shown 
in Figure 2 . If both the aiming and ballistic errors are 
normally distributed, the MPPs should be normally 
distributed, with a standard deviation given by 

o'mpi 2 = a a + — cr b 2 , (1) 

n 


where a a is the standard deviation of the aiming error, 
(7b is that of the ballistic error, and n is the number 
of projectiles per salvo. The value of ompi can be 
found by the use of probability paper, or arith- 
metically. 

7.2.2 An Example 

To show the details of calculation, the following 
table represents a series of 20 salvos of two bombs. 
The value of X\ and x 2 are the range errors of the 
bombs in feet. The third column gives the MPPs 
found by averaging the first two columns. The next 
two columns are the deviations of the individual 
bombs from the MPI. The 40 values of Ax were ar- 
ranged in ascending order of magnitude from — 26 to 
+26, and plotted on probability paper as shown in 
Figure 1, the lowest being plotted at F n = 4 V, and 
the highest being plotted at F n = -fr. The resulting 
curve is about as good an approximation to a straight 
line as we can expect from a sample this small, so it 
was assumed that the distribution was normal, and 
the best straight line drawn in by eye. From the 
points at F n = 0.5000 and F n = 0.8413, a is found to 
be 18 feet. Since n = 2 , the standard ballistic disper- 
sion Gb is 18 a/ 2 = 25 feet. The distribution of the 


CONFIDENTIAL 


134 


OPERATIONAL EXPERIMENTS WITH EQUIPMENT AND TACTICS 


MPFs was then plotted as shown in Figure 2. This 
also seems to be normal, with ompi = 161 feet. Hence 


( T a = ^ o"mpi 2 — -o& 2 = 160 feet. 

A similar calculation could have been made for the 
deflection errors. 


Record of deviations of hits from aiming point 


Salvo 

Xi 

x 2 

MPI 

Axi 

Ax 2 

1 

-124 

-142 

-133 

9 

-9 

2 

-68 

-82 

-75 

7 

-7 

3 

-52 

-88 

-70 

18 

-18 

4 

184 

162 

173 

11 

-11 

5 

-220 

-254 

-237 

17 

-17 

6 

-42 

-14 

-28 

-14 

14 

7 

218 

200 

209 

9 

-9 

8 

-6 

-20 

-13 

7 

-7 

9 

-52 

-78 

-65 

13 

-13 

10 

276 

264 

270 

6 

-6 

11 

-108 

-78 

-93 

-15 

15 

12 

48 

82 

65 

-17 

17 

13 

156 

128 

142 

14 

-14 

14 

-38 

-8 

-23 

-15 

15 

15 

-52 

-102 

-77 

25 

-25 

16 

-90 

-42 

-66 

-24 

24 

17 

-8 

-60 

-34 

26 

-26 

18 

-16 

-4 

-10 

-6 

6 

19 

66 

48 

57 

9 

-9 

20 

154 

154 

154 

0 

0 


7.3 EVALUATION OF DETECTION 
EQUIPMENT 

The field testing of detection equipment should be 
conducted to determine the numerical values of the 
measures of effectiveness of the equipment in the 
tactical situations in which the equipment is to be 
used. For some equipment the only important meas- 
ure to be determined is its sweep width against the 
most important target or targets it is designed to 
detect. In other cases, however, the sweep width is 
not sufficient. An early warning radar, for example, 
must not only detect aircraft, but do so at a long 
enough range to enable an interception to be made. 
In such a case it is necessary to know the “survival 
curve,” i.e., the probability of a plane approaching 
to a given range without being detected. In other 
cases the accuracy of range and bearing information 
is important and must be determined. 


7.3.1 Sweep Width 

From a theoretical standpoint the most direct 
method of determining a sweep width is from direct 
trial. For airborne radar such trials might be carried 
out by having the equipment flown a distance L in 
an area A containing n targets. If the flying and the 
placing of the targets are done perfectly at random, 
and C contacts are made, the sweep width W is given 
by 

W = — . (2) 

nL 

In actual practice, however, randomness is very diffi- 
cult to achieve, and to avoid difficulties with edge 
effects, the area A must be of dimensions large com- 
pared to W. This requires a great amount of flying 



to be done to make C large enough to avoid trouble 
with statistical fluctuations. Because of these diffi- 
culties the method of direct trial is very rarely used. 

The most common method of evaluating search 
equipment is what might be called the range distri- 
bution method. In this method the equipment is car- 
ried toward a target (or the target made to approach 
the equipment) until the target is detected, and the 
range of first detection is recorded. From a suffi- 
ciently large number of such runs, a distribution 
curve can be constructed showing the probability 
that the target has been detected as a function of 
range. A typical form for such a curve is shown in 
Figure 3. In many cases the probability of detection 
approaches unity as the range approaches zero, but 
this is by no means always the case. The only uni- 


CONFIDENTIAL 


SURVIVAL PROBLEMS IN GUNNERY 


135 


versal characteristic of such a curve is that the proba- 
bility is a monotonic decreasing function of range. 

7.3.2 Detection Expectancy 

As shown in Chapter 5, the probability of detection 
is related to the “detection expectancy,” 0, by the 
relation 

P = 1 - . (3) 


By means of this equation, a plot of 0 as a function 
of range can be constructed. In Figure 4 the <t> versus 



Figure 4. Variation of detection expectancy with range. 
If / is the slope of the curve, then — fdR is the probability 
of detecting a target between ranges R and R -f dR. 


R plot corresponding to the P versus R plot of Figure 
3 is shown. If P approaches unity as R approaches 
zero, then <f> approaches infinity. 

The negative slope, / = —d<f>/dR of the 0 versus R 
curve represents the probability of detecting a hither- 
to undetected target in an element of range dR, 
divided by dR. It is therefore a direct measure of the 
detecting power of the equipment at a given range. 
Examination of this quantity will frequently reveal 
“bad spots” in the performance of the equipment. 

In actual service the target motion is seldom 
straight toward the detection equipment. The most 
common situation is that in which the target tracks 
are straight lines passing the target at random. If the 
doubtful (but usually unavoidable) assumption is 
made that the detecting power 0 is independent of 
the target aspect and bearing, then the curve of de- 
tection probability as a function of the lateral range 
of the target track can be found. If the observed 
relationship between / and R can be approximated 


analytically, then for a track whose lateral range is x, 
the detection expectancy is given by the integral 

=J" AVx 2 + y 2 )dy , (4) 

and the probability of detection is as usual given by 

P(x) = 1 - e ~^ x) . (5) 

If no analytical expression for / can be found, or if 
the integral (4) proves difficult, then can be 
foun d graph ically by plotting a series of curves for 
+ y 2 ) as functions of y, and measuring their 
areas with a planimeter. The sweep width can then 
be found by 

W =j° P(x)dx. (6) 

Examples of such calculations can be found in Di- 
vision 6, Volume 2B. If the target aspect produces 
an effect on sighting, this can be introduced as an 
additional factor in the integrations. It is, of course, 
highly important to set up the experiments so any 
aspect effect will be apparent. 

7 4 SURVIVAL PROBLEMS IN 

GUNNERY 

A type of analysis very similar to that above has 
been used in gunfire evaluation, particularly the 
evaluation of A A fire. In the A A case the pertinent 
measure of effectiveness is the probability of shooting 
down a plane before it reaches a position to drop 
bombs or torpedoes. The situation is similar in other 
cases where the object of the gunfire is to prevent the 
approach of enemy forces. 

7.4.1 AA Gunnery 

The evaluation is based on practice firings in which 
a target (e.g., a drone) is made to approach the gun 
position, using evasive maneuvers similar to those 
which an actual enemy might be expected to em- 
ploy. Firing is continued at a known rate throughout 
the entire run. The range of each hit (or burst of 
proximity-fuzed ammunition) is recorded. The target 
need not be of the same size as the enemy craft 
against which the effectiveness is to be measured, 
but, if it is not the same, the ratio of the effective 


CONFIDENTIAL 


136 


OPERATIONAL EXPERIMENTS WITH EQUIPMENT AND TACTICS 


target areas must be known. Different types of run 
(varying altitude, speed, line of approach, etc.) 
should be tried, and the results of each type analyzed 
separately. 

As the firing rate and quality of the ammunition 
may vary from run to run, these must be recorded 
during the firing practice. In analyzing the results, 
a standard firing rate and quality of ammunition are 
chosen. If, in a given run, the firing rate was j times 
the standard, and the quality of the ammunition 
(fraction of shells not duds) was q times the standard, 
then the run is given a weight jq, that is, it is con- 
sidered that jq standard runs were made during the 
actual run. 

The first step in the analysis is to find the average 
number of hits per standard run outside each range, 
as a function of the range. This is conveniently done 
by arranging all the hits in order of decreasing range, 
and numbering them serially. The quotient of the 
serial number of each hit by the total number of 
standard runs then gives the average number of hits 
outside the range of the hit. The result is now plotted 
against range to give the hit expectancy curve for a 
standard run. 


firing. Moreover the hit expectancy is directly pro- 
portional to the effective target area, so that the 
curve is easily translated from the practice target to 
the actual target being considered. 

If E is the hit expectancy at range R, the proba- 
bility that the target will be hit before it reaches a 
given range is given by 

p = 1 - e ~ E . (7) 


In most cases, however, it is not this probability 
which is of interest, but the probability that the tar- 
get will be destroyed. To find this it is necessary to 
introduce the damage coefficients, D n (Chapter 6). 
To a sufficient degree of approximation the proba- 
bility of n hits, if E are expected, is given by the 
Poisson law : 


The probability of destruction is therefore 


Pd 


X®-* 


E n e~ E 

n\ 


(8) 


7.4.2 Hit Expectancy 

The hit expectancy curve is the analogue of the 
detection expectancy curve for the detection prob- 
lem. It should be proportional to the firing rate, so 
that the effect of a change in the number of guns or a 
change in the firing rate per gun is easily found. It is 
also directly proportional to the quality of the ammu- 
nition. Its negative slope is a measure of the effec- 
tiveness of the fire as a function of range, and can be 
used to find the weak spots of any given method of 


Given the D n , this may be evaluated as a function 
of E once and for all, and the resulting curve used to 
convert the hit expectancy curve into a probability 
of destruction curve. When the “vital spot” hypothe- 
sis can be used 

1 -D n = (1 -D)\ 

and 


P D = 1 - e~ DE . 


(9) 


In this equation DE is evidently the expected num- 
ber of times the plane will be destroyed. 


CONFIDENTIAL 


Chapter 8 

ORGANIZATIONAL AND PROCEDURAL PROBLEMS 


S ince operations research cannot work in a 
vacuum, the problems of organization and of re- 
lationship with the military are not trivial ones. As a 
matter of fact, in the last war a great deal of the time 
and energy was spent, by scientists and officers, in 
finding workable solutions to the organizational prob- 
lems involved in setting up operations research 
rather than in doing operations research itself. It 
took careful organizational planning and detailed in- 
doctrination of workers to insure that technical infor- 
mation could be sent freely across command bound- 
aries without short-circuiting the usual chain of 
command in regard to orders. It took a great deal of 
missionary work to persuade security officers that it 
was important to release highly secret information to 
the operations research worker, even though he did 
nothing more than think about the information for 
the time being. 

By the end of the war, most of these organizational 
problems had been worked out, in one way or an- 
other. Several possible types of organization had been 
evolved and the possible procedural methods for 
some of the specialized operations research work 
(such as the working up of operational statistics; or 
research in the field) had been determined. The 
present chapter will discuss some of these solutions 
and will indicate some of the problems requiring 
special consideration. 

8.1 ORGANIZATION OF AN 

OPERATIONS RESEARCH GROUP 

The activities of an operations research group are 
unfamiliar to the armed services, and many of the 
necessary procedures of such a group run directly 
contrary to long established precedents of military 
organization. Ordinarily, breadth of knowledge of a 
military situation, command responsibility, and 
power go hand-in-hand in the military organization. 
The soldier in the lower echelon is supposed to know 
just enough to get his own job done, and his power 
and responsibility are commensurate with his knowl- 
edge. The high command, on the other hand, has 
access to all of the information concerning the mili- 
tary situation, and concurrently has broad powers 
and responsibilities. It is a fundamental property of 


operations research that operations research groups 
must have broad knowledge, but should have very 
little power and responsibility. Operations research 
workers must be able to think about the military 
situation impersonally and impartially, and this can 
be done best if they are relieved as much as possible 
of the responsibility of issuing orders. Their conclu- 
sions must take the form of advice to some high- 
ranking officer, for him to make the orders (if he sees 
fit). 

Because of the close interconnection between 
breadth of knowledge and breadth of responsibility 
and power in military organizations, the principle of 
“normal channels” is particularly strong in such 
organizations. In order that the system work effec- 
tively in times of great stress, the system of hierarchy 
of power and responsibility must be clear-cut. Each 
officer must be answerable to only one superior, and 
the men under him must be answerable only to him, 
otherwise conflicting orders arise and the system falls 
to pieces. Since breadth of knowledge usually coin- 
cides with breadth of power and responsibility, it has 
been taken for granted that the channel for the trans- 
mission of intelligence must be identical with the 
channel for transmission of orders and requests. This 
communalization of the intelligence channels with 
the command channels is satisfactory as long as the 
intelligence is not overly technical or is not urgent. 
If the information to be transmitted from headquar- 
ters to the field or from the field to headquarters is 
both technical and urgent, however, experience in 
the last war indicates that the normal command 
channels are quite inadequate. They are too long, 
and the links in the chain usually consist of officers 
with little technical knowledge, and technical knowl- 
edge cannot he transmitted via nontechnical inter- 
mediaries. 

8.1.1 Importance of Contacts with Several 
Echelons 

Let us now see what implications these general 
comments have to bear on the problem of organizing 
an operations research group. Ideally, such a group 
should have available all possible information con- 
cerning a given type of warfare; the results of its 


CONFIDENTIAL 


137 


138 


ORGANIZATIONAL AND PROCEDURAL PROBLEMS 


work could be findings and recommendations on the 
conduct of all aspects of this type of warfare, from 
minor details of maintenance and training to overall 
strategical questions. Due to the usual hierarchy of 
responsibility, power, and knowledge, this output 
must be fed into the military organization at several 
different levels, depending on the level of the corre- 
sponding findings. The group, therefore, should have 
access to several different levels in the military 
hierarchy. 

An operations research group, moreover, cannot 
work insulated from direct experience. It probably is 
impossible, and it certainly would be inefficient, to 
have the group segregated in a single room or build- 
ing, with all its data and all requests for studies 
pushed through a slot from the outside. An extremely 
important part of the functions of an operations re- 
search group is to determine what are the important 
problems to be solved as well as in the solving of 
them. It is true in operations research, as it is in other 
parts of science, that the proper enunciation of the 
problem to be solved often requires a higher order of 
scientific ability than does the solving of the prob- 
lem, once it has been formulated. It is not to be ex- 
pected that nontechnical officers, immersed in the 
pressures of command responsibilities, should be 
able to formulate the problems for the operations 
research group to work on effectively; such a division 
of labor would drastically reduce the group’s useful- 
ness. The operations research worker himself must 
get close enough to the action to be able to help 
formulate the problem as well as to work on its 
solution. 

Both in the interests of rapid formulation of the 
problems and rapid dissemination of the solutions, 
therefore, it is important that the operations research 
group have contact at a number of different echelons 
in the military hierarchy. This inevitably means the 
cutting across of command boundaries. The expe- 
rience of the last war has shown that operations re- 
search can only function effectively and adequately 
by and through such cross connections. The problem 
of the supplementing of “normal channels” is there- 
fore a fundamental one in the organization of an 
operations research group. In fact, one can say that 
it is a waste of valuable technical talent to form an 
operations research group without having worked 
out a solution of this problem which is satisfactory 
both to the scientists and to the military officers 
involved. 


8.1.2 Assignment of Group 

It must also be apparent that an operations re- 
search group must be attached to the operational 
commands in a military organization. The logistic 
and technical commands of the organization also 
have their problems requiring scientific personnel for 
their solution, but this sort of work is not what is 
meant by operations research. The operations re- 
search worker must be the scientific adviser of the 
fighting force itself, and must never degenerate into 
a salesman for a laboratory or a service branch. He 
must be able to render impartial judgments on va- 
rious equipment, so as to pick the one most effective 
for the operation at hand, not the one which happens 
to be urged by a bureau. It is important, not only 
that the operations research worker feel that he is 
part of the fighting team (even to the extent of being 
somewhat suspicious of bureaus and laboratories), 
but also that the operational command be thoroughly 
persuaded that the worker is really a member of the 
fighting team. The operations research group should 
therefore be assigned directly to the operating com- 
mand, and should make its reports and recommenda- 
tions directly to the various echelons in this com- 
mand. 

8.1.3 Sub-Groups in the Field 

It should be clear from the foregoing that an oper- 
ations research group should be attached to a high 
echelon in the headquarters staff, but should also 
have points of contact provided with lower echelons 
in the field. With the U. S. Navy this was achieved 
by assigning the group as a whole to the Readiness 
Division of the Headquarters of the Commander-in- 
Chief, U. S. Fleet, and by reassigning parts of the 
group to the strategic planning officer, or the opera- 
tions officer, of subordinate commands in the field. 
The central group at headquarters was answerable 
only to headquarters, and distributed its reports only 
on headquarters authorization. The sub-groups, how- 
ever, were assigned to their respective theater com- 
mands, and distributed their reports only with the 
authorization of these commands. Thus contact was 
made with several different echelons in the military 
hierarchy. 

Intercommunication between the sub-group and 
the headquarters group was carried on directly and 
frequently, with the approval of headquarters and of 


CONFIDENTIAL 


ORGANIZATION OF AN OPERATIONS RESEARCH GROUP 


139 


the theater commands. Toward the end of the war, 
biweekly teletype conferen ces were held between the 
central group and the sub-groups at Pearl Harbor, 
■with a resulting improvement in this imp ortant inter- 
communication. Formal reports sent from the central 
group to the sub-group were subject to the approval 
of headquarters; similarly, reports sent from the sub- 
group back to headquarters were subject to the ap- 
proval of the ‘local theater commander. Informal 
communication also went on, giving facts, not gossip 
or personal opinions. Information gained by such 
intercommunication was not disseminated outside 
the group until permission had been obtained from 
the source. Thus, with a well-indoctrinated group, 
technical data could be obtained rapidly without dis- 
turbing the normal channels for command. 

It was also found important to circulate the per- 
sonnel between field assignment and headquarters 
group. Much information can only be “soaked up” in 
person in the field, but such information can often be 
most useful back in headquarters. Conversely, the 
headquarters worker is likely to lose touch with 
reality unless he is “sent to the front” once in a while. 
A period of rotation of about six months seems to be 
healthy. Such rotation often is opposed by the field 
commands, particularly when they have a good man 
assigned them, but the rotation is important enough 
for the homogeneity of the group and for the alert- 
ness of its individuals, to risk local displeasure by 
maintaining the rotation. 

8.1.4 Reports and Memoranda 

The output of an operations research group con- 
sists of reports and memoranda; the reports embody- 
ing the results of major studies, and the memoranda 
consisting of comments on various aspects of the 
changing military situations and of suggestions for 
action. The reports generally come from the central 
or headquarters group, where the members have the 
leisure and facilities to carry out long-term studies. 
The sub-groups at the outlying bases usually apply 
the results of these basic studies, and their output is 
more likely to consist of shorter notes and memo- 
randa. All of this output must be scrutinized carefully 
to see that the material is consistent with other pre- 
vious reports and with latest group opinion, as well as 
to see that the material does not conflict with current 
military doctrines or criticize unduly certain military 
actions. 

Allegorically, the group might be considered to be 


separated from the rest of the military organization 
by a semipermeable membrane. From each one of its 
contacts with the military organization, at head- 
quarters and at the theater commands, data concern- 
ing military operations is absorbed through the mem- 
brane into the group. Inside the group these data and 
related ideas, suggestions, criticisms, and theories 
should circulate freely. Internal memoranda and 
studies, which are not supposed to circulate outside 
the group, should promote this rapid flow of ideas 
and suggestions. Only in this way can the proper 
atmosphere of freedom of thought be built up, with- 
out which no scientific advance can be made in the 
field. 

Because of the great difference between military 
procedure and scientific procedure, however, this 
interplay of suggestion, criticism, and theory should 
be kept within the group until the new ideas and con- 
cepts have crystallized. A military organization, by 
nature, finds it difficult to understand such an inter- 
play; and a broadcast of the procedure to the service 
at large only produces misunderstanding and suspi- 
cion. Therefore, the semipermeable membrane must 
be so designed that it will allow material to go from 
the group out to the service at large only if this ma- 
terial has come to represent the considered opinion of 
the group as a whole, on the basis of all available 
data at that time. 

Consequently, the written material of the group 
must be of two sorts: internal studies, representing 
preliminary theories and suggestions, which circulate 
freely within the group and to the sub-groups, but 
which are not distributed to the rest of the service 
except under very special circumstances; and the 
reports and memoranda mentioned above, which are 
written primarily for circulation to the rest of the 
service. These reports and memoranda must be care- 
fully edited and refereed. They must embody the 
considered opinion of the group as a whole, and 
should not contradict previous reports (or if they do 
so contradict, an explanation should be included as 
to why the group has changed its mind) . They must 
he brief. They should be written so as to be under- 
standable and easy to read for the average opera- 
tional officer. These officers are usually overworked 
and should not be expected to take the time to ab- 
sorb complicated arguments or unclear writing. Great 
effort must be made by the group to make the major 
points easy to understand and the reasons simple to 
grasp in all reports and memoranda written for dis- 
tribution outside the group. 


CONFIDENTIAL 


140 


ORGANIZATIONAL AND PROCEDURAL PROBLEMS 


8.1.5 Status of Group Members 

In order to emphasize the difference in function 
between the operations research worker and the staff 
officer, it has usually been found best to leave the 
operations research worker in a civilian status, at 
least within the continental United States. In an 
outlying theater of operations, however, it is usually 
necessary for the worker to be in uniform. Sometimes 
the worker has been given a temporary rank, suffi- 
cient for him to perform his functions without undue 
embarrassment. This temporary rank has some dis- 
advantages, however, for it immobilizes him in the 
military hierarchy, and makes it more difficult for 
him to approach lower echelons on terms of equality. 
Sometimes it has been possible to avoid the question 
of temporary rank, and give the worker some special 
insignia. This also has difficulties, for proper accom- 
modations and entrance into necessary headquarters 
are often only available to officers, and the special 
insignia may not be recognized as being the equiva- 
lent of an officer. 

Most of these problems are individual ones, and 
no general solution can be offered. More comments 
on the status of the worker in the field will be given 
in the next section. 

8.1.6 Recruiting and Training the Operations 

Research Worker 

No particular correlation has been found between 
the particular scientific specialty in which the opera- 
tions research worker was trained and his subsequent 
excellence in the field of operations research. Since 
the subject is a new one, no scientists have been 
trained primarily in operations research, and it is 
unlikely that such specialized training will become 
prevalent soon. Consequently, the majority of the 
workers in operations research must be recruited 
from men trained in other branches of science. It is 
obvious that their previous training should be in 
science, since operations research is a branch of 
science; but it is not certain which science would pro- 
vide the best training. In fact, it appears that each 
science has its own usefulness in the training. 

Mathematicians are perhaps the most useful in 
headquarters groups, particularly if they have train- 
ing in the field of probability (which does not mean 
the usual course in statistics). Their capacity for ab- 
stract thought makes them particularly valuable in 
analytical studies although an interest in practical 


applications seems to be more useful than a bent 
toward complete abstraction. 

Men with training in physics are also particularly 
valuable, since most of the weapons of war operate on 
principles well known to physicists. A great number 
of the evaluations of equipment performance must be 
made by men with training in physics or electrical 
engineering, and physicists are useful in helping de- 
sign operational experiments. Training in chemistry, 
particularly in physical chemistry, also has proved to 
be a good background. Training in biology also seems 
to be quite useful, since the data of operational re- 
search often has the same refractory quality as the 
data of biology. Psychologists and economists also 
have their special spheres of usefulness. 

It has been found to be rather futile to attempt to 
train operations research workers in special courses 
away from an operations research group, though this 
may be possible in the future. It is useful to train the 
future worker in the properties of military equipment 
and in military doctrine, but he must learn opera- 
tions research by doing operations research. During 
the last war there was not accumulated enough basic 
data in this field to serve as object material for a class 
to work on. In addition, problems of organization and 
contact with the military changed so from time to 
time, and from group to group, that it was almost 
more important to learn the workings of some par- 
ticular group and how it was allowed to function 
than it was to learn details of the subject ahead of 
time. Consequently, men with solid training in basic 
science, with a flair for research and an interest in 
theory, were chosen, and the apprentice method of 
training was used instead of the classroom method. 

Perhaps in the future, as operations research comes 
to be better recognized and has accumulated a back- 
ground of experience and knowledge, it will be ad- 
visable to give courses in the subject for prospective 
workers in this field. 

8.2 OPERATIONS RESEARCH IN THE 
FIELD 

Activities of representatives in the field form an 
important part of operations research. This has been 
found so in all the groups devoted to this type of 
work, including the U. S. Navy and Army, in their 
several branches, and the British. This section will 
attempt to formulate and illustrate some of the 
aspects of field work. The examples will be drawn 
from the experience of the operations research group 


CONFIDENTIAL 


OPERATIONS RESEARCH IN THE FIELD 


141 


[ORG] assigned to and functioning with the Head- 
quarters of the Commander-in-Chief, U. S. Fleet; it 
is felt, however, that they are probably general in 
nature. 

Work of the field representatives serves several 
purposes: (1) to provide help directly to the service 
units to which they may be temporarily attached; 
(2) to secure information that might otherwise be 
difficult to obtain, and transmit it to the parent 
operations research body; (3) to give the individual 
members of the operations research group a practical 
background which is indispensable in avoiding the 
pitfalls to which the pure theorist may be subject. 

The value of maintaining representatives with 
operating military units throughout the course of an 
operations research group’s activities is thus made 
apparent. 

8.2.1 Assignments 

The principal types of field assignment include : 

1. Liaison. 

2. Staff. 

3. Operations. 

4. Training. 

5. Experimental (operational). 

6. Experimental (equipment). 

Representatives of all types are desirable in the 

field. In actual practice, it is seldom that a field man 
is restricted to work in any one of these types; indeed, 
his assignment is likely to represent a mixture. 

The liaison type is best illustrated by the repre- 
sentatives of ORG in London, where their primary 
mission was to secure information for the parent 
body, and indirectly for the Navy. This information 
was obtained from British operations research groups 
and from the British military services. The ORG rep- 
resentatives also kept the British units posted on the 
work of the ORG home office (to the extent permitted 
by the naval staff) and thus helped to minimize a 
duplication of research. 

In the staff type , the representative is assigned di- 
rectly to the staff of a commander of some operations 
unit, such as a sea frontier, a fleet, an area or the like. 
This representative acts as a small operations re- 
search group in his own right, seeking solutions to 
problems appropriate to his duties that are proposed 
to him by the staff commander. In addition, his loca- 
tion at headquarters proves extremely useful in the 
collection and systemization of operations statistics 
and in forwarding them to the central ORG. This 


type of assignment has been found very fruitful, and 
has been used often both in Atlantic antisubmarine 
work and in the Pacific. 

Perhaps the best example of the staff type of work 
is found in a Pearl Harbor sub-group, consisting of 
several men assigned to Commander Submarines, 
Pacific. This sub-group conducted the bulk of its 
work at the command headquarters, getting into the 
“field” and participating in operations in connection 
with such projects as required such activity. 

In most staff type assignments, such as to sea fron- 
tiers, the men have also engaged in operations type 
activities, going on operational missions to observe 
the techniques involved. This contact with lower- 
echelon operating personnel was used to disseminate 
directly such technical information as seemed desir- 
able. Thus, a representative at a sea frontier could 
spend a considerable portion of his time visiting va- 
rious bases where he studied the local records, lec- 
tured and talked informally with the personnel, and 
gained personal knowledge of the technicalities of the 
operations. Purely operations type assignments have 
been infrequent; perhaps the closest approximation 
to these were assignments to an aircraft carrier escort 
[CVE] during an antisubmarine warfare [ASW] 
cruise, to a submarine on war patrol, and to a task 
force during a landing operation. 

The training type assignment has also been infre- 
quent, but some excellent work has been done by 
men assigned to such training bases as at Langley 
Field, Virginia, in connection with Army ASW work, 
and at Kaneohe, with Fleet Air Wing 2. On such as- 
signments, studies were made of improving operators’ 
techniques through training, with an eye to the best 
methods. At such locations it was also frequently 
possible to find approximations to operational data 
by studying the results of training for the operation 
under study, which were quite useful when it hap- 
pened to be difficult to obtain operational results. For 
example, a study of ASW bombing errors from opera- 
tional data was difficult because of the scarcity of 
well-recorded data, while data on training-bombing 
errors were available in statistically significant quan- 
tity. 

An example of the experimental ( operational ) type 
of work was to be found in the assignment of mem- 
bers to the Antisubmarine Development Detach- 
ment, Atlantic Fleet [ASDevLant], the one official 
unit which made experimental studies of proposed 
operational techniques in the field of antisubmarine 
warfare. Here the operations research representa- 


CONFIDENTIAL 


142 


ORGANIZATIONAL AND PROCEDURAL PROBLEMS 


tives contributed importantly, through their study 
of the results of operational tests. A somewhat simi- 
lar assignment was at Langley Field, to the Army 
Air Forces development unit for antisubmarine 
work. In general, it is regarded as desirable to main- 
tain one or more operations research representatives 
at stations devoting their efforts to experimental 
operational study. 

The sixth type, experimental ( equipment ), is not 
properly part of an operations research group’s 
work. Of necessity, however, it has been found fre- 
quently that this type of work can be done expe- 
diently by the group’s men, because of location or 
other reasons. Such work has been carried on at 
ASDevLant, at Langley Field and, toward the end 
of the war, at the naval ordnance test station 
[NOTS] atlnyokern. 

All six types of work might typically be included 
at a single base. A man assigned to a sea frontier, in 
addition to staff duties, would observe operational 
work in the field, function as liaison between his sta- 
tion and the headquarters group, might make some 
studies of training results, and might occasionally 
carry out experimentation on a minor scale at the 
request of the commanding officer. 

Of the six types, a combination of the staff and 
operations types has been found most fruitful in pro- 
viding assistance to the commands and information 
to the central group. While each type has obvious 
merit within its defined scope, the assignment to the 
staff of the commander of a fairly large operating 
unit has proved of greatest value, and should prob- 
ably constitute the bulk of the field work in a future 
war. 

Assignment of field men to the Trinidad Sector of 
the Caribbean Sea Frontier is cited as an excellent 
example. They reported directly to the commander 
of the sector, and thus may be regarded as having a 
staff assignment. There they were available for con- 
sultation on planning and the results of operations. 
As in most broadly organized bases, there was ample 
variety of operations to require of the men a certain 
amount of the five other types of work. Proximity to 
operating squadrons enabled the men to keep in 
close contact with the field. Studies were made of the 
training results in these same squadrons. Liaison was 
carried on, both in forwarding operational needs to 
Washington and in explaining doctrine developed in 
Washington to the operating forces. At various times, 
experimental work was done, for instance on the 
study of the effects of German search receivers, and 


of the use of American search receivers in searching 
for German radar. 

8 2.2 Types of Field Work 

An operations research man on field assignment 
encounters various types of work, which fall into 
these primary divisions : 

1. Analytical. 

2. Statistical. 

3. Liaison. 

4. Experimental. 

5. Educational. 

6. Publication. 

The analytical type of problem consists of the 
study of an operation, major or minor, before its 
execution, or at least before data on the results of the 
operation are available for examination. Examples of 
this are numerous: the design of aircraft barrier pa- 
trols in the Fourth Fleet Area to intercept subma- 
rines, or the study of antitorpedo evasive maneuvers 
by the Submarine Operations Research Group, 
SubPac. 

An operation can also be studied by a statistical ex- 
amination of the operational results. As typical ex- 
amples of the numerous studies of this sort carried on 
during the war may be mentioned the examination 
of contacts to determine the probable use of search re- 
ceivers against radar, or studies of the relative hazards 
to convoy and independent merchant vessels. A more 
complete discussion of specialized statistical tech- 
niques is given in the next section. 

Liaison work has been discussed to some extent 
above and hardly needs amplifying here. This work 
is a natural necessity, and a certain amount of it 
must be done at each base. 

A certain amount of experimental work is a neces- 
sary and proper function of an operations research 
field man. This does not mean that it is his proper 
duty to indulge in the design or development of new 
equipment, but he can play an important role in 
studying the use of new equipment or new opera- 
tions. When a new piece of equipment, such as a 
radar set or a bombsight, is first available for field 
use, it is desirable to assign, if possible, the first 
output to units operating in an area serviced by an 
operations research man. Then the initial use of the 
equipment can be scientifically observed and sug- 
gestions formulated to aid in introducing the inno- 
vation elsewhere in the field. Similarly, when a new 
operational technique is provided, such as the use of 


CONFIDENTIAL 


OPERATIONS RESEARCH IN THE FIELD 


143 


flares in a night aircraft attack on a submarine or 
surface vessel, it is again desirable to try it out under 
careful observation and to make suggestions for the 
elimination of imperfections which may appear. 

The educational aspect of field work is twofold: 
education of the operations research man in the 
methods of the field, and such instruction work in 
the field as he is able to provide. The first, self-edu- 
cational, aspect is of extreme importance to the suc- 
cess of an operations research group, and should be 
encouraged at every opportunity. It is only by gain- 
ing intimate practical acquaintance with operations 
problems that successful operations research can be 
done. 

The educational contributions to the service per- 
sonnel which are within the sphere of activity of the 
field representative include both formal and informal 
instruction. It is not unusual for him to be called 
upon to deliver a lecture or series of lectures to oper- 
ational squadron personnel either at a school or at 
their own base. Informally, he has frequent contacts 
with operational personnel, both professionally and 
socially, which present opportunities for fruitful dis- 
cussions. 

The field man is occasionally called upon to aid in 
the publication of professional periodicals issued by 
the command to which he is attached. Examples of 
this are the Statistical Summary published monthly 
by the Trinidad Sector of the Caribbean Sea Fron- 
tier, the Submarine Bulletin published by the Com- 
mander Submarines, Pacific Fleet, and the monthly 
Antisubmarine Bulletin af the Seventh Fleet. Rep- 
resentatives of the Antisubmarine Warfare Opera- 
tions Research Group have had important roles in 
the formation and continued publication of these 
bulletins. This type of work is regarded as most valu- 
able, inasmuch as it acquaints a wide circle of readers 
with statistical and analytical studies. 

A different method of subdividing operations re- 
search work would be into strategic and tactical 
work. A great deal of the operations research in the 
recent war, but by no means all, has been in the tac- 
tical field, and this has been even more true of field 
men. If a field man, however, is attached to a suffi- 
ciently high echelon command, he may be called upon 
to study strategic problems. For example, an ORG 
man at Argentia worked on the question of routing 
convoys, and ORG men at London worked on the 
problems of optimum size of convoys and the Bay 
of Biscay antisubmarine offensive. 


8.2.3 General Comments 

The following general comments concerning field 
assignments are made on the basis of the experience 
of the ORG during the past war. It is to be remem- 
bered that, while these suggestions probably would 
apply to other cases, situations might well develop 
which would demand entirely different treatment. 

Selection of the location for a new field assignment, 
and the initial installation of a man, depends upon 
several factors. Considerations apart from opera- 
tions research determine the location of some types, 
such as liaison, training, or research, and such as- 
signments are almost automatic. On the other hand, 
the location of a combined staff-operations type 
assignment requires careful consideration. Selection 
of regions of extensive activity in the operations 
under study should be made with considerable care. 

It must be understood that assignment is made 
only upon direct invitation of the officer to whom the 
man will be ordered. This requires formal or informal 
machinery to inform the proper officials of the exis- 
tence and availability of operations research men and 
of the type of service they can perform. 

Assignment to as high a command as possible, 
compatible with the area and type of problem under 
study, is generally desirable. This permits the widest 
distribution of the field man’s work. 

In reporting to a new assignment, the field man 
should try to take with him a carefully chosen li- 
brary of material bearing on his proposed work. 
Usually the parent body can supply a wider variety 
of literature than is available in the local command. 
This library should be kept up to date during the 
assignment. 

After being assigned, a field man works on prob- 
lems presented to him by the commanding officer of 
the staff as well as on problems originating with him- 
self. The ultimate decision as to the work to be done 
remains with the commanding officer. The field man 
should reach an agreement with this officer that, as a 
scientist, he should have complete access to the data 
pertaining to an assigned problem. He is to make all 
official reports to the same officer, with such distribu- 
tion as the latter approves. However, informal com- 
munication with ORG headquarters should be ar- 
ranged for, as this is highly desirable for the inter- 
change of ideas. These informal communications are 
generally shown to the officer in charge, for his infor- 
mation rather than for censorship. 


CONFIDENTIAL 


144 


ORGANIZATIONAL AND PROCEDURAL PROBLEMS 


Provision for opportunities for frequent trips about 
the operational area and to ORG headquarters at 
least every six months should be arranged, to stimu- 
late further the exchange of ideas. 

The field man is enjoined to remember that these 
assignments should be terminated as well as started. 
Once he is well established in a given location, nat- 
ural inertia may be counted upon to slow the process 
of terminating the assignment and completing the 
work in that field. It should be understood that war is 
a constantly changing operation, and assignments 
must be shifted to keep field men in appropriately 
active regions. The supply of such men is limited, 
and should not be wasted in dead-end streets. 

Experience has proved that a field man can fre- 
quently function best when in uniform. At a location 
where almost all personnel are in uniform, the pres- 
ence of a civilian often presents a problem. Local 
rules and regulations are often not designed to handle 
such a situation. The British met the problem by 
commissioning operations research men as officers 
for the period of their field assignments. The ORG of 
the Commander-in-Chief, U. S. Navy [COMINCH, 
CJSN] designed a uniform of its own for its field 
men out of the country. This was a modified officer’s 
uniform, and was intended to reflect a professional 
and social status for field men equivalent to officers. 
Attempts have been made in the past to have the 
field men work in civilian clothes, but, despite cer- 
tain advantages, the disadvantages in an active 
theater of war are serious. The British have felt that 
there were security difficulties arising from a charac- 
teristic uniform reflecting the nature of the work 
done by operations research men, particularly in 
active theaters. For field men operating in such areas 
as the continental United States, where the presence 
of civilians is normal, and provision for handling 
civilians is available, the necessity for a uniform does 
not apply. 

In the last analysis, the success or failure of a field 
assignment is fundamentally determined by the per- 
sonality of the man involved. His background should 
include a certain amount of physics, engineering and 
mathematics, including statistical theory. He should, 
moreover, be well trained in most phases of his war 
work. And above all, he should have a personality 
that will permit him to talk successfully to all ranks, 
from the bottom to the top, as the measure of his 
achievement may depend on this basic ability to 
adapt himself to all grades of military personnel. 


8.3 STATISTICAL METHODS IN 
OPERATIONS RESEARCH 

Operations research has as a large part of its field 
the analysis of past operations. The methods em- 
ployed to carry out these analyses are usually statis- 
tical in nature with the following general purposes in 
mind: 

1. To measure the overall effect of past operations 
and to determine and measure the effect of the va- 
rious factors that have significantly influenced the 
overall result. For example, the total number of 
enemy submarines sunk or damaged by aircraft dur- 
ing a particular period of time is the final measure of 
the absolute effectiveness of aircraft as an antisub- 
marine weapon during that time. A proper compari- 
son of the various circumstances of aircraft attacks on 
submarines will indicate the relative influence of each 
individual circumstance on the final result. Suppose 
that it is desired to determine the effect of the speed 
of the aircraft on its success in attacking a submarine. 
A simple comparison of the proportion of successes 
attained according to the speed employed during the 
attack run will provide an approximate answer to the 
problem. However, in using such an elementary pro- 
cedure we must guard against the possibility that the 
results are affected by other factors which vary as 
maximum plane speed varies. For example, different 
types of planes, in addition to having different maxi- 
mum speeds, may drop different size bomb sticks or 
have other different qualities. 

It immediately becomes apparent that the more 
precise the designation of the variable to be analyzed 
is, the more difficult is the problem. One direct 
answer to this difficulty would be to tackle the prob- 
lem by the statistical method of measuring partial 
correlation. This, however, involves in essence a 
mathematical formulation of the variation of the 
different factors, often a major task in itself. One 
alternative is to compute an “expected” result for the 
particular operation under analysis by taking into 
account existing knowledge of the effect of variables 
other than the one being analyzed. In the example of 
aircraft attack speeds, the comparative effect of the 
length and spacing of bomb sticks may be obtained 
from theoretical studies. This knowledge, along with 
similar knowledge or assumptions for the other fac- 
tors involved in the problem, may in turn be used to 
compute the “expected” number of submarines sunk 
or damaged, independent of any consideration of 


CONFIDENTIAL 


STATISTICAL METHODS IN OPERATIONAL RESEARCH 


145 


speed. Then a comparison of the actual to the “ex- 
pected” or computed results within each of the speed 
classes will indicate the effect of speed. Another al- 
ternative is to select a number of attacks within 
which all of the circumstances except speed are 
exactly the same. That is to say, variation in the type 
of plane, the size of the bomb stick, the depth of the 
submarine, and in all other characteristics of the 
attack with the exception of speed is eliminated. The 
effect of speed on the success of the attack can then 
readily be ascertained. Unfortunately, the volume of 
operational data is seldom, if ever, great enough to 
permit this type of analysis. 

It is obvious that the second method is the most 
practical means of analysis, and it has the further 
advantage of combining laboratory and field tests 
with operational data. 

2. To estimate the effects of future operations on 
the basis of past experience, where future operations 
depend upon changes in tactics, in weapons, or in 
conditions. This might be considered the ultimate 
aim of all statistical analysis in operations research. 
For instance, in the above example the result will 
lead directly to an estimation of the increase in the 
number of submarines which are likely to be sunk or 
damaged if the attack speed is increased. Another 
type of problem might involve the use of a weapon 
which has never been used before. Here operational 
experience and laboratory expectations must be com- 
bined to produce an estimate of the absolute and rela- 
tive effect of such a weapon. 

8.3.1 Planning a Procedure to Handle the 
Statistical Problem 

This section points out the importance of planning 
a statistical job before it is started, and outlines the 
elements which should be considered. The following 
sections treat the details of some of the more impor- 
tant elements. 

The scope of the job itself should be clearly defined, 
and limited to existing and immediate problems. 
Each statistical item recorded should serve a very 
definite purpose, and the number of such items should 
be limited to fit the manpower assignment and the 
time fixed to complete the job. This indicates a fair 
amount of advance planning. There is always a ten- 
dency to attempt to do too much when starting a new 
task. The reasons for this are twofold: often the pur- 
pose of the statistical analysis is too indefinite, re- 
sulting in a tendency to record everything which 


might be of interest; and, secondly, the feeling that 
if reports must be read it takes little more time to 
record a maximum amount of data as compared to a 
minimum amount. This type of mistake may result 
in applying a disproportionate amount of time to 
recording data with the result that the analyses are 
delayed. Tactics change rapidly in wartime. Thus the 
usefulness of an analysis is apt to be vitiated by de- 
lays, regardless of the improved quality of the slower 
analysis. 

It is particularly important to realize that changes 
in tactics or emphasis are apt to require a change in 
the kind of data to be recorded. This is another rea- 
son for confining the records to immediate problems. 
New problems should be taken up as they arise. 

Examples of these difficulties may be found in a 
number of cases where records were set up to record 
significant data from action reports of a particular 
type of warfare. One such example may be found in 
the experience of the Submarine Operations Research 
Group, U. S. Navy. After considerable discussion 
and planning, a comprehensive system of punch 
cards was set up to cover the significant phases of a 
submarine patrol. The primary idea was to have a set 
of records which would take care of as many as pos- 
sible of the problems which might arise in the future, 
as well as the immediate problems then demanding 
attention. All the data for these records, five differ- 
ent kinds of cards, were to be obtained from one read- 
ing of the submarine war patrol reports. The five 
kinds of cards were as follows: 

1. Area Patrol Card — A summary of the targets 
sighted, targets attacked, results of the attacks; 
number, type and results of counterattacks, in each 
patrol area; time in area. 

2. Submarine Incident Card — A detailed record of 
the conditions under which each target was sighted. 

3. Torpedo and Gun Salvo Card — A detailed rec- 
ord of each torpedo salvo fired at a target, giving the 
situation of the target and of the submarine, the 
type of torpedoes or gun fired, the settings and firing 
conditions. 

4. A/C Contact Card — Record of each sighting of 
an enemy aircraft and the events resulting from the 
sighting. 

5. Counterattack Card — A detailed record of the 
situation of the submarine and counterattacking 
craft and the results of such counterattacks. 

Apparently there was little left to be desired in this 
set of records. However, the following difficulties soon 
became apparent : 


CONFIDENTIAL 


146 


ORGANIZATIONAL AND PROCEDURAL PROBLEMS 


1. There was a great deal of manpower tied up in 
recording the data, a considerable amount of which 
would never be used. 

2. The data recorded did not fulfill the require- 
ments of some of the problems which had arisen after 
the system was set up, so that the reports had to be 
reread anyway, with a definite problem in mind. 

3. Rereading the reports indicated that, if the sys- 
tem were to be continued, many changes w T ould be 
necessary, due to changes in points of emphasis and 
interest. It was then decided to discontinue recording 
most data which were not being used at the time for a 
definite purpose, resulting in important simplifica- 
tions of the original punch cards. 

It is not always possible to avoid the difficulties 
pointed out in the above. Research always implies a 
certain amount of groping, either to find a problem 
or to find the solution to a problem. But looking for 
problems need not involve more than an intelligent 
reading of a representative sample of the reports, 
while looking for a solution may require considerably 
more data than can be used, although it often can 
be confined to fairly definite limits. In starting a sys- 
tem it is generally better to use simple methods of 
analysis confined to one problem at a time, until the 
limitations of the basic information are known. Often 
a solution to a problem may be obtained from an 
analysis of a sample of the total data available. The 
size of the sample should, of course, be such as to give 
significant results. In some cases, when a new prob- 
lem arises, the necessary data may be obtained from 
the reports coming in during a certain period after 
that time. This avoids backtracking. The principal 
idea is to avoid investing time in nebulous future 
problems. 

8.3.2 Mechanics of Collecting and Recording 
Data 

Before starting a statistical job the source of data 
should be checked for accuracy and completeness. 
This may be done by determining by what means and 
for what purpose the data were originally gathered, 
and by reading through a sample of the material. The 
consistency of action reports varies considerably, de- 
pending upon the preciseness with which the format 
is prescribed. Some such reports are almost useless 
insofar as statistical analysis is concerned, while 
others are prepared on prescribed forms especially 
designed to obtain data for analysis. An example of 
the latter form are the antisubmarine action reports. 


Often it is necessary to begin the analysis by arrang- 
ing for the data to be collected. 

Reports concerning operations have a bewildering 
variety of forms and usually occupy a sizeable frac- 
tion of the free time of the operating forces of many 
different commands and agencies; reporting on the 
use of material (complete histories of each torpedo 
are kept, for instance, which return to the torpedo 
test station after expenditure of the torpedo); re- 
porting on specific aspects of an operation (forms are 
made out for each aircraft mission, which are turned 
in to the local command) ; reports on personnel, which 
go to type commands or to training commands; sum- 
maries of action, which eventually get to headquar- 
ters, etc. Any further forms to be made out usually 
are opposed by the operating forces, on the valid 
grounds that already too many are required (in fact, 
a promising field for operations research would be the 
study of report forms, with a view to reducing their 
complexity) . 

Therefore, it is important to make a thorough 
investigation of all reports (even to sending workers 
to subordinate commands to look over reports which 
never get to headquarters) before recommending ad- 
ditional reports to be made out. If it is clear that no 
existing report contains the data, and if it is impor- 
tant that the data be obtained, then the new report 
form should be made as simple to fill out as possible. 
After designing the form, it should be tried on a num- 
ber of persons acquainted with the operation (but not 
with details of the analysis to be made) to see whether 
the instructions and questions are clear, and can be 
given unambiguous answers. 

Every possible occasion should be made to acquaint 
the operating forces with the fact that their action 
reports are important and are used, rather than just 
being filed. 

The next step is to select a type of card for record- 
ing the required data. Various types of cards are dis- 
cussed in the following paragraphs. 

8.3.3 IBM (Hollerith) Cards 

These are punch cards. The particular position of a 
single hole in a column represents a number and the 
particular positions of two holes in a column repre- 
sent a letter. Information punched into the card is 
interpreted at the top of the card. (See Figure 1.) 
This type of card admits of the use of high speed 
automatic machinery to sort, tabulate and record the 
data in the cards. 


CONFIDENTIAL 


STATISTICAL METHODS IN OPERATIONAL RESEARCH 


147 


B 2 1 4 


1 T T 1 

10 

i I??. 14 


'6 H 18 I9!20|21 22 23 24 

27 28 kUIiI C 33 >lU 

38 3 7 38 3t]40|4l 42 43 44] 

WSM* 47 48 49 

»b. » M * 

| 

lw 57 58 5b] 

m 

161 62 63 84' 

cL « 68 89| 

[70|71 72 73 74ll5[76 77 78 79|l» 

































1 2 3 


5_ 5 


8 9 10 

11 

12 13 « 

15 



1 

27 28 29 

20 

l 31 32 33 

1 34 35 38 37 

38 39 40 

41 

f_i3_M 

45 

46 47 48 

i 49 50 51 52 

53 54 55 56 57 56 59 

*• 

1 2 


4 5 

6 

7 8 

9 

10 11 


2 13 u 


15 ie 

? 

ie 19 

2C 

21 22 

23 24 

) 26 27 2 

8 

29 20 

3! 

32 33 

3' 

1 35 36 


7 38 39 

4C 41 4 

\Z 

43 44 

45 

0 0 0 0 

0 

0 0 0 0 

0 

0 0 0 0 

0 

0 0 0 0 

0 

0 0 0 0 

0 

0 0 0 0 

0 

0 0 0 0 

0 

0|0| 

0 

1010 

B 

ofloi 

0 

gooo 

0 

0|00 

0 

0 0 0 0 

0 

0 0 0 0 

0 

0 0 0 0 

0 

0 0 0 0 

0 

1111 

1 

1111 

1 

1111 

1 

mg 

1 

1111 

1 

1111 

1 

1111 

1 

1111 

1 

1111 

1 

mi 

1 

1111 

1 

11J1 

1 

1111 

1 

1111 

1 

1111 

1 

1111 

1 

2 2 1 2 

2 

2 2 2 2 

2 

2 2 2 2 

2 

2222 

2 

1 2 2 2 

2 

2 2 2 2 

2 

2 2 2 2 

2 

2 1 2 2 

2 

2 2 2 2 

2 

2 2 2 2 

2 

2 2 2 2 

2 

2 2 2 1 

2 

2 2 2 2 

2 

2 2 2 2 

2 

2 2 2 2 

2 

2 2 2 2 

2 

3 3 3 3 

s 

3 3 3 3 

3 

3 3 3 3 

3 

3333 

3 

3 3 1 3 

3 

3 3 3 3 

3 

3 3 3 3 

3 

3 3 3 1 

3 

3 3 3 3 

3 

3 3 3 3 

3 

3 3 3 3 

3 

3 3 3 3 

1 

3 3 3 3 

3 

3 3 3 3 

3 

3 3 3 3 

3 

3 3 3 3 

3 

4 4 4 4 

4 

4|44 

4 

4 4 4 4 

4 

4 4 4 4 

4 

4 4 4 4 

B 

4 4 4 4 

4 

4 4 4 4 

4 

4 4 4 4 

4 

1444 

4 

4 4 4 4 

4 

4 4 4 4 

4 

4 4 4 4 

4 

1444 

4 

4 4 4 4 

4 

4 4 4 4 

4 

4 4 4 4 

4 

5 5 5 5 

5 

5 5 5 fi 

5. 

5 5 5 5 

I 

5 5 5 5 

5 

5 5 5 5 

5 

5 R 5 5 

5 

5 5 5 5 

5 

5 5 5 5 

5 

5 515 

5 

5 5 5 5 

5 

5 5 5 5 

5 

5 5 5 5 

5 

5155 

5 

5 5 5 5 

5 

5 5 5 5 

5 

5 5 5 5 

5 

O 

1 

2 " 

3 

" 4 

5 

‘ 6 

7 

TSTb 

b 

mi 

o 

fflTFS 


mi 

6 

6bbb 

6 

6 6 Sg 

t 

6 6 6 6 

8 

*66T 

6 

6 6 6 6 

S 

6 6 6 6 

6 

6 6 6 6 

6 

6 6 6 6 

6 

b b|l 

6 

6 6 6 6 

6 

6 6 6 6 

6 

6 6 6 6 

6 

7 7 7 7 

7 

7 7 7 7 

7 

7 7 1 7 

7 

7777 

7 

7 7 7 7 

7 

7 7 7 7 

7 

J777 

7 

7 7 7 7 

7 

7 7 7 7 

7 

7|77 

7 

7 7 7 7 

7 

7 7 7 7 

7 

7 7 7 | 

7 

7 7 7 7 

7 

7 7 7 7 

7 

7 7 7 7 

7 

8 8 8 8 

8 

8 8 8 8 

8 

8 8 8 8 

I 

8 8 8 8 

8 

8 8 8 8 

8 

8 8 8 8 

8 

8 8 1 8 

8 

8 8 8 8 

8 

3 8 8 8 

8 3 8 8 | 

8 

3 8 8 8 

8 

8 8 8 8 

8 

8 8 8 8 

i 

8 8 8 8 

8 

8 8 8 8 

8 

8 8 8 8 

8 

9 9 9 9 

9 

9 9 9 9 

9 

9 9 9 9 

9 

9199 

9 

9 9 9 9 

9 

9 9 9 9 

9 

9 9 9 9 

I 

9 9 9 9 

9 

9 9 9 9 

9 9 9 9 9 

9 

B 9 9 9 

9 

9 9 9 9 

9 

9 9 9 9 

9 

9 9 9 9 

9 

9 9 9 9 

9 

9 9 9 9 

9 

1 1 2 3 4 

s 

17 1-1 

10 

11 12 1) H 

1i 

16 T7 11 19 

20 

21 22 23 24 

25 

28 27 28 29 

30 

31 32 33 34 


38 37 38 39 

40 

41 42 43 44 

45 48 47* 48 49 

50 

51 52 53 54 

» 

51 57 58 59 

GOIol 92 S3 84)65 

9 67 68 69 

78 

71 72 73 74 

751 

78 77 78 7* 

80 


IBM 5282 IIMNUO FOR U$£ UNDER PMMT 1./72.482 IBM SERVICE BUREAU 


Figure 1. Sample Hollerith or IBM punch card, with data punched. 


Ordinarily, two steps are involved in transcribing 
information from the action reports to the punch 
cards. The first step involves picking the required 
data from the report and translating them into a 
coded form. A transcript card (see Figure 2) is invalu- 



Figure 2. Sample of transcript card to collect data 
from action report for later punching or transcribing. 


able in performing this step. The second step involves 
punching up a punch card on a machine, which is 
operated in much the same manner as a typewriter, 
from the coded data on the transcript card. A con- 
siderable amount of hand labor is involved in this 
procedure, ordinarily more than is required for an 
ordinary written record. However, the great advan- 
tage of the system comes from the subsequent opera- 
tions of sorting, tabulating, and recording results. 
Consequently, the key for determining the advan- 
tage of an IBM system, over a system of ledger cards 
for instance, is to weigh the additional work involved 


in transcribing against the time saved in the subse- 
quent operations. It is quite evident that the savings 
in cost is more or less proportional to the amount of 
analytical work involving sorting and tabulating 
which is required of a set of data. 

The code for translating data to the IBM cards is 
the most important item in the entire procedure of 
planning the statistical job. As a sample a part of the 
code used in the analysis of antisubmarine attacks 
by aircraft is given below. 

Column 

No. Code 

1 Assessment 

1 Submarine known sunk 
. 2 Submarine probably sunk 

3 Probably seriously damaged (A) 

4 Probably seriously damaged (B) 

5 Probably slightly damaged, 
etc. 

2-5 Incident number (identifies incident) 

6-9 Coordinated attack information. A separate card is 
made up for each attacking air or surface craft. Col. 
6 identifies the various cards by a symbol A, B, C, 
etc. Col. 7 indicates total number of attacking sur- 
face craft; Col. 8, total number of attacking aircraft 
and Col. 9 indicates percentage of credit to the par- 
ticular craft represented by the card. 

10 Nationality or Branch of Service 

1 Navy, heavier than air 

2 Navy, lighter than air 

A Coast Guard 

J Marine Corps, 
etc. 


CONFIDENTIAL 


148 


ORGANIZATIONAL AND PROCEDURAL PROBLEMS 


Column 

No. Code 

11-12 Type of own aircraft 

OA PB4Y 

OB PB2Y 

1A PBY 

IB PBM, etc. 

O indicates very long range bomber, 1 indicates long 
range bomber, etc., and the second symbol further 
identifies the plane as to particular type. 

13-16 Month, day, year 

17 Day or night (i.e., degree of natural light during 
attack) 

1 Day 

2 Morning twilight 

3 Evening twilight 

4 Night (moonlight), 
etc. 

18-22 Position in whole degrees 

39-40 Range of radar contact in miles 

41 Type of radar in plane 
P band 

1 ASVC, SCR, 521, ASE, Mk II 

2 P or L band, unknown type 
L band 

3 ASA 

4 ASD 
etc. 

44 Degree of submergence of U-boat at time of release 
of bombs 

0 Fully surfaced 

1 Decks awash 

2 Stern or conning tower 

3 Periscope 

4 Down 0-15 seconds, etc. 

The code is the link between the raw data and the 
statistical card. It also represents a compromise be- 
tween the limitations of the former and the require- 
ments of the latter. Since the final classification of 
the data can be no better than the classification of 
the code, great care should be exercised in making up 
the code. It should not call for a finer classification 
of data than is obtainable from the reports, or indi- 
cate a better quality of data than actually exists. This 
often happens from a tendency to classify arbitrarily 
an item when its class is unknown. It is better to have 
a code to catch the unknown items. Occasionally an 
item may be reported in a way that was not antici- 
pated in the code. It may be proper to throw such 
an item in with some existing class or to set up a code 
for a new class. In either event the code should reflect 
such a decision. For instance, in the above code for 
degree of submergence, if it were reported that the 


U-boat was “diving” it might be proper to consider 
for the purposes of the study that “diving” repre- 
sented about the same as “stern or conning tower,” 
in which case the code should be amended to read 
“stern, conning tower or diving.” 

8.3.4 Ledger and Dual Purpose Cards 

The principal objections to punch cards are as 
follows: 

1. A punch card, being mostly in coded form, is 
not very legible for quick reference. 

2. Its limited size does not always permit a suffi- 
ciently complete description of the incident for the 
purpose of the statistical job. 

3. There is no convenient way of recording unusual 
events or background events often important in some 
types of analysis. 

4. Classification of varied descriptive matter may 
be too complex to work with. 

A handwritten card meets all these objections; 
however, of course, it suffers seriously by comparison 
when it comes to analyzing the recorded data. Con- 
sequently, handwritten cards are not recommended 
unless the job is small, or contains data not subject 
to very much analysis, or is used in an auxiliary ca- 
pacity. It is sometimes possible to combine the IBM 
transcript card with a written record, so as to retain 
some advantages of both systems. (See Figure 3.) 

8 3.5 Key Sort Cards 

A key sort card is an ordinary written card con- 
taining all the required information in a written 
form, and, in addition, there are holes near the edges 
which, when punched out singly or in combination, 
represent a code. The number of items which may be 
coded and punched, however, cannot exceed much 
more than a dozen. Sorting is done in the following 
manner: 

We will refer to a sample key sort card (Figure 4) 
designed to record data on merchant vessel losses. 
Suppose we wish to punch the section of the card 
labeled “Own damage” in upper left-hand column. 
We might have a code as follows: 1, ships known lost; 
2, ships overdue, presumed lost; 3, ships damaged, 
etc. If for a particular case the code were “1,” then 
the hole over “1” would be slotted out; if the code 
were “3,” then the holes over “1,” “2,” and “SF” 
would be slotted out. Sorting is then done by thrust- 


CONFIDENTIAL 


STATISTICAL METHODS IN OPERATIONAL RESEARCH 


149 



Estimote for Incident: S ,P ,D_ 

Killed 8efore Altock 

WEAPONS DROPPED 

BOMBS WT Lb», no. 

WT. L b», NO. 

TORP 


POSITION FAILURES 


Figure 3. 


Record card to supplement punched card, giving details not desirable to punch. 


ing a needle successively through the various holes. 
All the cards which fall out are those wanted. The 
“SF” hole identifies a double punch. 

Key sort cards represent an attempt to capture 
some of the best features of the punch card and the 
written card. Their use may be justified in a number 
of instances. In cases where a written card is desir- 
able or necessary and the amount of sorting and 
tabulating of data is not great, the key sort card 
may give a real advantage over either the written 
or punch card. In general, the same considerations 
outlined for IBM coding will apply for key sort 
coding. 

8 - 3 - 6 Mechanics of Analysis 

Before beginning the job of preparing the data for 
study, an outline should indicate the combination of 
variables to be studied. For instance, the combina- 
tion “Type of own aircraft,” “Day or night” and 
“Assessment” would indicate how various types of 
aircraft fared under various conditions of natural 
light. It should be pointed out, however, that, what- 
ever analysis is planned at this point, it should have 


been anticipated as far as possible before any part of 
the job was started. 

It is of great importance to have some form of con- 
trol data. These may be similar figures for a pre- 
vious roughly equal period of time, or it may be the 
expected results explained near the beginning of this 
chapter. If expected results are to be used a proba- 
bility of success may be computed separately for each 
action, taking into consideration the actual condi- 
tions of the action and any other knowledge, theo- 
retical or operational, necessary to determine the 
expected probability of success. This result may be 
written on or punched into the card and tabulated 
along with the other figures. 

At this point a brief description is given of the 
various IBM machines which may be used for the 
mechanical analyses of IBM cards. 

8.3.7 IBM (Hollerith) Machines 

Alphabetic Key Punch — This machine punches the 
cards with either numerical or alphabetical data. 
The operator presses a key for each number or letter 


CONFIDENTIAL 


150 


ORGANIZATIONAL AND PROCEDURAL PROBLEMS 



Figure 4. Sample key sort card for uniform collection of data and simple mechanical manipulation. 


to be punched in much the same way as a typewriter 
is operated. Depending upon the skill of the operator 
and the type of code used, up to 1,000 cards may be 
punched per day. 

Alphabetic Verifier — Verifies the punching of the 
cards. It is operated in about the same manner as the 
key punch machine. 

Alphabetic Interpreter — Prints selected informa- 
tion from the punch card on either or both of two 
lines at the top of the card. It reads and prints about 
60 cards per minute. 

The three machines described briefly above are 
necessary to prepare the punch cards from the ab- 
stract cards, which had been prepared from the raw 
data. Machines described below are used in the me- 
chanical analysis of the data. 

Card-Counting Sorter — Reads cards at the rate of 
400 per minute. It will sort cards into any desired 


order (either numeric or alphabetic) on one or more 
columns. For any one column at a time it will count, 
for the entire file of cards, with or without simulta- 
neous sorting, the numbers represented by the posi- 
tion of the punch in the column. That is, if a hole is 
punched in the “4” position in a selected column on 
one card and in the “6” position in the same column 
on a second card, the counting device will register 
“1” in the 4 counter and “1” in the 6 counter for 
these two cards. The machine will also select these 
cards from the file in one run. The machine will 
also select from the file, in one run, all those cards 
having a predetermined punching in any field of 
columns. 

The counting device on this machine performs one 
of the functions of the Alphabetic Accounting Ma- 
chine described below, but in a very limited sense, as 
will be seen. 


CONFIDENTIAL 


STATISTICAL METHODS IN OPERATIONAL RESEARCH 


151 


Alphabetic Accounting Machine {commonly called 
the 11 Tabulator”) — Performs the following functions: 

1. Listing. Prints all or a selected amount of infor- 
mation from each card at the rate of 80 cards per 
minute. 

2. Tabulating. Adds or subtracts the figures in all 
or a selected number of columns and, if desired, lists 
cards at the same time. If not listing, tabulating is 
done at the rate of 150 cards per minute. Totals may 
be printed at any break in the control variable. For 
instance, if the variables X, Y, and Z describe a class 
and the cards have been sorted so that Y is under X 
and Z is under Y, the machine may be made to print 
a total of the figures in particular fields, as well as the 
number of cards going through the machine, when- 
ever the values of X, Y, or Z change. This gives class 
totals. The machine in addition will give totals from 
separate counters when only X or Y changes, and 
when only X changes. 

In addition to the above the machine may be set 
up so that whenever a card punched with a code, for 
example X, in a particular column goes through 
the machine an additional operation will be per- 
formed. This additional operation may be anything 
within the normal capacity of the machine, such as 
listing, adding or subtracting into designated count- 
ters, printing totals, etc. 

Multiplier — Has a card speed of from 500 to 1,500 
cards per hour, depending upon the number of digits 
involved in the multiplications. The machine will 
multiply two figures of up to 8 digits each and punch 
the answer to any desired number of places in the 
card. It will perform many types of operations in- 
volving multiplication, addition, subtraction, and 
crossfooting of two or more factors. 

This machine and the Tabulator may be used to 
make lengthy calculations as are involved in deter- 
mining standard deviation, correlation coefficient, 
etc. 

Reproducing Summary Punch — Operates at the 
rate of 100 cards per minute. This machine will re- 
produce a file of cards in toto or in part, or it may be 
wired to reproduce only a part of each card, or to re- 
produce a part or all of each card simultaneously 
changing the position of the columns on the card. It 
may be used to gang punch predetermined informa- 
tion onto all or part of a file of cards. Either repro- 
ducing, or gang punching, or both, may be done se- 
lectively on the basis of a characteristic punch on the 


cards. This machine may be connected to the Tabu- 
lator for the purpose of punching a summary card 
each time the Tabulator prints, a total, which card 
may be punched with certain group characteristic 
information together with the totals which have 
been accumulated by the Tabulator. 

Collator — Operates at a maximum rate of 480 
cards per minute. It may be used for merging two 
files of cards, for withdrawing from a file cards with 
predetermined punching in a maximum of 32 col- 
umns, or withdrawing from the file all cards with 
numerical punching between two limits, as, for ex- 
ample, the selecting of cards representing antisub- 
marine actions between two limits of latitude and 
two limits of longitude. 

8 . 3.8 Study of Data and Preparation of 
Final Report 

After the data are prepared in tabular form they 
must be examined for significant variation. Trends in 
the data may be brought out by graphically fitting a 
smooth curve to points. The forces behind these 
trends should be examined so as to obtain a reason- 
able explanation for the variation. Later on a more 
satisfactory result may be obtained by fitting the 
data, by least squares or some other method, to a 
mathematical curve. If this is done the various fac- 
tors in the formula representing the curve should 
bear physical explanations. 

Before preparing the final report, a decision should 
be made as to the type of report which is required. 
This refers mostly to the amount of reference data 
to be included. A pure reference report is simply a 
compilation of data for all the important variables 
in the form of tables and graphs without any con- 
clusions as to significance of any variation or trend in 
the data. This is certainly not operations research. 
Any data presented should be for a definite problem. 
The writer of the report should be willing, and in a 
position, to make responsible conclusions in a form 
which the operations officer to whom the report is 
directed can use directly in formulating doctrine or 
making decisions regarding plans or doctrine. He 
should present the data in such a way as to convince 
the reader of the correctness of his conclusions. How 
this is to be done will depend upon the importance 
of the conclusions, the type of problem, and perhaps 
the individual preferences of the reader. 


CONFIDENTIAL 







TABLES 


153 



Table I. Random sequence of digits. 


34829 

12006 

51850 

01054 

55930 

Derived from Tippett, Random 

Sampling Numbers, 

09655 

44407 

72675 

10410 

22229 

Cambridge University Press. 192 

!7. (See Section 2.1, 

29974 

68015 

40277 

55815 

90984 

Table 1 of ChaDter 2. and Section 6.4.) 


52490 

09053 

35850 

32398 

53650 






06538 

69698 

49007 

23532 

38896 

57705 

13094 

60835 

36014 

35950 

33690 

45040 

45744 

98683 

27307 

71618 

35193 

42323 

38612 

03235 

61708 

87590 

96911 

60166 

15298 

73710 

64560 

25732 

93857 

73606 

36074 

10144 

60456 

97834 

33252 

70131 

64559 

93364 

33749 

66090 

56365 

76064 

61446 

05141 

83928 

16961 

68008 

63407 

08921 

31842 

88288 

18915 

01484 

42971 

23435 

53324 

39848 

72028 

07721 

22807 

65421 

84885 

58127 

84117 

12627 

43166 

33851 

25496 

58577 

41476 

67798 

74145 

14569 

48861 

30367 

26275 

80586 

83761 

39303 

74473 

69184 

26754 

16211 

09156 

23333 

05926 

69939 

58568 

19302 

78489 

59886 

21304 

99988 

01241 

60360 

66289 

98351 

27409 

17068 

14142 

90654 

37385 

53211 

34771 

69359 

35483 

32673 

64789 

59201 

75975 

69921 

03959 

66049 

65690 

45320 

09393 

12949 

78992 

18688 

55604 

57959 

76536 

71359 

53990 

73195 

30304 

14644 

67388 

73449 

80702 

63609 

56656 

02834 

56855 

50876 

55186 

66887 

75316 

41734 

11027 

98462 

84131 

30962 

16608 

32627 

64003 

43042 

73673 

17033 

34559 

63578 

50128 

71712 

70101 

70556 

20514 

49110 

21681 

18664 

73345 

92500 

97454 

32388 

46167 

93500 

00188 

18170 

32763 

94722 

02783 

03289 

01648 

69834 

55659 

16023 

55709 

19187 

50983 

55024 

54095 

95414 

93649 

22686 

26319 

20078 

86977 

02464 

98359 

85143 

29373 

30781 

56663 

67117 

05596 

92740 

31303 

55739 

38440 

28594 

96006 

28838 

07809 

64571 

68116 

43583 

11578 

52992 

78142 

76086 

69351 

31502 

60309 

74417 

30379 

76145 

93045 

86513 

25730 

97570 

07995 

93655 

23699 

59729 

24354 

69477 

93011 

10480 

30454 

26292 

00900 

91757 

82052 

60805 

51929 

34027 

42844 

56437 

19106 

07120 

29396 

17901 

85317 

91288 

13396 

23426 

52906 

13647 

58222 

11851 

17727 

05558 

09792 

41101 

59527 

70231 

09461 

57910 

45818 

24806 

25424 

31648 

65813 

70855 

25154 

61461 

99602 

54062 

96748 

90506 

38695 

08183 

90183 

34523 

06277 

72870 

69962 

23767 

45732 

39116 

02624 

04408 

57276 

35855 

08366 

85109 

31311 

43191 

91542 

35745 

36522 

97706 

72549 

42280 

46410 

32961 

27004 

03283 

78115 

82713 

56461 

14925 

83202 

50674 

19199 

75530 

65339 

46250 

18186 

07938 

62250 

00994 

07555 

82548 

10664 

91269 

93382 

28366 

61450 

51275 

73071 

43496 

32443 

70353 

89865 

63191 

05758 

16074 

74582 

32203 

59362 

03051 

66017 

06433 

65211 

52786 

00336 

98951 

80604 

51925 

98178 

11444 

47321 

05789 

61382 

06016 

88222 

54686 

49538 

24693 

40526 

06864 

35326 

82795 

41116 

95670 

98585 

87615 

22917 

16837 

74412 

85993 

08367 

49336 

45915 

25019 

52103 

44968 

99135 

78155 

79033 

92473 

64742 

94781 

65421 

65952 

91827 

27709 

58274 

97412 

62192 

75906 

66400 

01912 

77234 

77311 

07069 

59560 

01940 

09892 

96942 

98187 

66808 

01330 

23430 

93999 

13928 

00799 

87397 

84299 

34623 

13830 

49703 

12138 

11171 

14363 

66674 

76151 

84445 

96036 

48259 

50518 

48680 

87881 

78118 

88178 

99279 

61716 

86012 

48472 

12634 

85445 

08606 

40057 

26616 

25825 

24202 

59298 

51625 

42687 

93997 

19323 

18011 

62325 

48154 

15821 

94010 

89923 

71881 

89434 

32799 

10514 

36580 

21041 

33802 

53561 

60981 

20327 

64466 

67912 

04011 

55961 

12790 

66413 

32462 

27047 

75626 

90180 

89489 

50359 

98156 

29560 

95580 

08834 

43767 

15735 

77367 

28023 

62721 

97152 

23640 

37295 

07771 

01100 

23449 

30044 

62046 

54017 

91319 

03727 

45005 

20270 

11113 

74904 

89595 

22767 

69759 

22721 

36524 

16443 

76193 

03878 

27161 

62190 

00088 

79130 

70600 

23292 

90567 

86755 

54159 

73133 

90578 

95308 

70756 

90830 


CONFIDENTIAL 


154 


TABLES 


Table I — ( Continued ) 


56656 

74058 

79714 

33580 

49320 

28995 

31970 

39110 

71164 

74969 

83635 

46160 

46799 

71120 

63317 

91719 

58831 

70122 

47229 

59911 

60112 

45549 

23508 

47961 

96059 

90722 

25293 

85013 

45087 

20376 

17557 

37568 

04058 

01851 

88070 

30266 

86373 

19069 

22246 

62328 

78543 

37058 

89772 

27159 

46158 

20980 

41329 

04998 

88478 

94797 

76762 

95282 

46440 

01328 

66375 

23542 

80262 

02462 

71287 

29772 

99512 

12069 

49701 

18345 

72659 

67839 

06339 

28240 

69027 

11868 

64594 

96862 

16958 

50558 

89049 

59559 

89230 

39081 

56032 

50874 

42957 

47491 

28028 

39521 

07309 

69161 

27766 

43883 

40266 

38074 

19660 

24412 

57042 

34137 

91103 

17323 

47446 

61047 

81370 

31174 

85144 

01109 

14180 

06475 

51450 

14284 

40876 

42047 

09332 

02897 

99106 

17378 

20375 

13116 

01359 

48956 

51457 

41422 

81782 

98690 

02438 

23725 

58885 

29126 

88618 

10712 

00829 

08793 

77802 

39699 

94918 

07866 

19220 

09379 

71119 

37052 

31862 

96291 

28953 

97538 

99343 

78243 

72801 

24279 

41276 

76178 

37668 

53493 

62114 

34825 

19748 

59795 

38219 

49232 

10148 

36708 

24299 

45132 

92481 

44499 

96428 

85178 

70368 

91546 

29881 

42829 

18331 

96411 

21303 

36397 

78388 

21440 

53420 

31554 

83349 


19268 

73248 

81305 

07701 

19415 

14078 

90001 

88821 

56944 

92014 

75825 

16401 

20301 

70654 

46231 

42705 

14426 

12088 

90910 

86701 

45709 

49315 

63916 

59272 

37804 

62509 

19795 

24528 

56275 

09077 

35964 

55927 

28935 

99538 

20821 

98225 

46074 

42975 

90799 

44421 

89995 

16835 

68290 

92188 

25552 

47202 

25233 

21637 

51827 

14084 

66414 

93439 

00550 

65926 

85223 

27671 

90798 

56370 

22726 

83527 

67780 

99540 

27914 

23458 

97548 

58418 

91310 

94209 

07136 

84089 

71261 

27258 

35317 

04121 

53460 

56061 

65853 

87865 

01757 

99359 

46275 

81178 

58592 

93289 

05065 

37907 

60879 

78292 

48656 

17171 

28329 

81037 

10070 

56832 

50895 

82877 

17434 

77931 

56768 

38316 

54306 

49594 

30946 

56677 

09312 

12803 

54403 

13160 

62353 

82501 

74173 

40297 

52041 

19082 

15843 

39539 

89717 

45036 

60698 

46286 

70937 

18423 

20323 

97672 

82885 

06256 

41832 

39121 

94175 

09627 

61618 

92582 

04569 

28897 

51244 

83223 

23505 

22187 

44185 

64919 

34877 

69215 

51377 

86341 

55073 

27107 

96364 

25416 

66339 

40116 

09312 

15960 

06929 

96272 

41919 

82501 

62062 

72868 

49368 

77275 

15843 

33995 

51332 

53150 

59071 

46286 

11459 

43599 

73166 

00626 

82885 

43910 

37202 

21139 

10557 


CONFIDENTIAL 


TABLES 


155 


Table II. Random sequence of angles, from 000 to 359. 
Derived from Table I. (See Sections 2.1 and 6.4.) 


143 

243 

012 

184 

359 

243 

015 

343 

094 

312 

307 

034 

083 

125 

211 

051 

160 

036 

115 

106 

001 

218 

004 

325 

211 

168 

164 

307 

346 

144 

070 

002 

130 

191 

106 

322 

043 

087 

317 

268 

044 

095 

032 

335 

319 

089 

245 

350 

246 

078 

158 

025 

246 

018 

316 

104 

343 

070 

342 

051 

348 

061 

276 

130 

309 

251 

017 

147 

066 

178 

258 

316 

037 

127 

174 

337 

192 

319 

303 

073 

205 

183 

171 

032 

117 

246 

120 

352 

156 

223 

057 

225 

014 

001 

277 

263 

343 

346 

259 

133 

250 

088 

205 

045 

007 

083 

037 

263 

011 

306 

313 

091 

272 

180 

028 

193 

017 

146 

019 

343 

270 

272 

058 

045 

127 

285 

143 

052 

042 

202 

262 

034 

299 

241 

222 

012 

220 

036 

213 

112 

059 

108 

018 

296 

274 

194 

259 

334 

219 

270 

115 

009 

267 

342 

135 

186 

036 

287 

309 

275 

336 

067 

294 

096 

049 

010 

140 

090 

278 

173 

179 

111 

164 

025 

288 

303 

293 

333 

027 

187 

081 

269 

304 

002 

359 

323 

341 

274 

102 

052 

169 

149 

329 

021 

012 

166 

159 

342 

000 

244 

315 

116 

254 

050 

280 

170 

171 

225 

159 

284 

354 

051 

048 

011 

074 

098 

126 

089 

120 

225 

149 

106 

018 

089 

359 

248 

103 

262 

299 

266 

003 

210 

331 

260 

115 

186 

311 

358 

123 

261 

303 

085 

309 

136 

103 

101 

135 

279 

056 

275 

242 

216 

232 

278 

186 

164 

277 

055 

219 

323 

094 

291 

200 

334 

101 

062 

120 

082 

040 

286 

139 

258 

319 

273 

323 

170 

035 

325 

106 

107 

093 

144 

328 

290 

049 

262 

213 

278 

089 

071 

096 

194 

030 

030 

142 

357 

102 

102 

061 

061 


319 

031 

059 

178 

232 

221 

262 

031 

280 

345 

008 

000 

142 

165 

120 

302 

174 

290 

350 

332 

144 

317 

104 

022 

246 

115 

197 

155 

151 

292 

086 

103 

036 

248 

339 

339 

324 

018 

204 

009 

199 

156 

055 

227 

275 

149 

242 

170 

204 

118 

063 

272 

246 

001 

272 

144 

083 

080 

297 

128 

258 

174 

286 

293 

121 

041 

078 

020 

104 

033 

252 

109 

130 

071 

272 

010 

339 

323 

007 

340 

127 

303 

349 

069 

136 

262 

271 

290 

342 

221 

214 

210 

094 

343 

211 

122 

183 

261 

035 

156 

126 

204 

244 

064 

133 

156 

154 

226 

299 

040 

338 

007 

273 

347 

219 

147 

118 

008 

060 

266 

114 

002 

237 

319 

254 

194 

159 

088 

288 

018 

332 

092 

075 

125 

037 

156 

328 

010 

065 

026 

116 

045 

120 

269 

359 

121 

289 

106 

055 

258 

166 

112 

168 

255 

110 

234 

164 

090 

316 

041 

171 

110 

164 

356 

316 

092 

027 

231 

270 

025 

341 

117 

184 

012 

191 

006 

288 

278 

032 

280 

116 

069 

353 

232 

012 

053 

087 

159 

313 

167 

185 

159 

159 

268 

213 

163 

345 

292 

114 

035 

320 

055 

159 

262 

083 

323 

252 

137 

026 

023 

079 

235 

083 

219 

028 

116 

111 

116 

119 

170 

089 

077 

062 

296 

076 

256 

318 

134 

056 

175 

191 

118 

333 

023 

182 

329 

177 

030 

031 

303 

250 

133 

202 

323 

275 

295 

272 

032 

253 

137 

191 

298 

274 

184 

337 

161 

114 

279 

000 

188 

347 

079 

323 

222 

269 

173 

180 

218 

023 

287 

091 

348 

212 

007 

134 

011 

253 

163 

062 

279 

071 

192 

251 

262 

170 

164 

077 

232 

277 

082 

037 

027 

098 

280 

206 

235 

277 

124 

247 

150 


CONFIDENTIAL 


156 


TABLES 


Table III. Random normal deviates in units of standard errors. 

Based upon tables of E. L. Dodd, Bollettino di Matematica, 1942, pages 76-77. (See equation (26) of Chapter 2 

and Section 6.4.) 

X 

X 2 

X 

X 2 

X 

X 2 

X 

X 2 

X 

X 2 


0.80 

0.640 

-0.69 

0.476 

0.38 

0.144 

0.13 

0.017 

1.73 

2.993 


-0.54 

0.292 

-0.21 

0.044 

-0.60 

0.360 

-1.59 

2.528 

-0.60 

0.360 


0.42 

0.176 

1.67 

2.789 

0.67 

0.449 

0.06 

0.004 

1.37 

1.877 

0.75 

-0.48 

0.230 

-1.13 

1.277 

0.50 

0.250 

-0.19 

0.036 

1.18 

1.392 


0.16 

0.026 

-0.61 

0.372 

0.74 

0.548 

1.16 

1.346 

0.37 

0.137 


1.95 

3.803 

1.57 

2.465 

-1.19 

1.416 

-1.47 

2.161 

0.35 

0.122 


1.87 

3.497 

1.41 

1.988 

-0.37 

0.137 

-0.25 

0.062 

-0.25 

0.063 


0.63 

0.397 

1.17 

1.369 

0.25 

0.062 

-0.24 

0.058 

-0.31 

0.096 

1.11 

-1.48 

2.190 

0.46 

0.212 

-1.28 

1.638 

-1.36 

1.850 

-0.83 

0.689 


-0.49 

0.240 

-0.27 

0.073 

1.04 

1.082 

-1.41 

1.988 

0.38 

0.144 


-2.92 

8.526 

1.53 

2.341 

-0.51 

0.260 

-1.02 

1.040 

-0.78 

0.608 


1.72 

2.958 

-0.08 

0.006 

1.29 

1.664 

-0.96 

0.922 

0.91 

0.828 


-0.90 

0.810 

-1.75 

3.063 

0.15 

0.023 

-1.09 

1.188 

-0.12 

0.014 

1.06 

-0.24 

0.058 

1.16 

1.346 

0.21 

0.044 

-0.22 

0.048 

1.23 

1.513 


0.24 

0.058 

0.16 

0.026 

0.28 

0.078 

0.75 

0.563 

0.96 

0.922 


0.34 

0.116 

0.06 

0.004 

-0.72 

0.518 

0.44 

0.194 

-2.27 

5.153 


-0.88 

0.774 

0.14 

0.020 

0.89 

0.792 

-0.14 

0.020 

-0.39 

0.152 


-1.07 

1.145 

0.54 

0.292 

-0.46 

0.212 

0.81 

0.656 

1.16 

1.346 

0.80 

0.47 

0.221 

-0.25 

0.063 

-0.01 

0.000 

0.59 

0.348 

0.56 

0.314 


1.46 

2.132 

-1.53 

2.341 

1.51 

2.280 

0.54 

0.292 

0.71 

0.504 


-0.67 

0.449 

-2.01 

4.040 

-0.52 

0.270 

0.67 

0.449 

0.05 

0.003 


0.61 

0.372 

-0.70 

0.490 

1.04 

1.082 

-2.01 

4.040 

-0.91 

0.828 


1.15 

1.322 

2.08 

4.326 

0.60 

0.360 

0.81 

0.656 

-0.77 

0.593 


-0.19 

0.036 

-0.95 

0.903 

0.56 

0.314 

-0.29 

0.084 

-0.22 

0.048 


-0.90 

0.810 

1.93 

3.725 

-0.57 

0.325 

-0.61 

0.372 

-1.61 

2.592 

1.07 

-0.70 

0.490 

-0.97 

0.941 

1.36 

1.850 

-0.02 

0.000 

0.87 

0.757 


-0.36 

0.130 

1.38 

1.904 

-1.24 

1.538 

-0.68 

0.462 

-0.92 

0.846 


0.05 

0.003 

-1.08 

1.166 

-0.49 

0.240 

-0.29 

0.084 

0.81 

0.656 


0.56 

0.314 

0.45 

0.202 

-0.37 

0.137 

0.26 

0.068 

2.37 

5.617 


1.28 

1.638 

1.25 

1.563 

1.34 

1.796 

0.83 

0.689 

-0.52 

0.270 


-1.18 

1.392 

-0.28 

0.078 

-1.23 

1.513 

-0.91 

0.828 

0.31 

0.096 


-0.66 

0.436 

-0.08 

0.006 

-0.76 

0.578 

0.75 

0.563 

1.75 

3.062 


-0.68 

0.462 

0.78 

0.608 

-0.96 

0.922 

0.15 

0.023 

1.78 

3.168 


1.76 

3.098 

0.39 

0.152 

-0.74 

0.548 

0.57 

0.325 

-0.80 

0.640 


-2.47 

6.101 

1.35 

1.823 

-0.33 

0.109 

1.66 

2.756 

0.75 

0.562 

1.13 

-0.32 

0.102 

-0.48 

0.230 

0.91 

0.828 

-1.99 

3.960 

-0.81 

0.656 


2.22 

4.928 

-0.22 

0.048 

-1.11 

1.232 

0.77 

0.593 

0.01 

0.000 


0.02 

0.000 

-0.35 

0.123 

-1.06 

1.124 

0.19 

0.036 

-1.59 

2.528 


-0.55 

0.303 

0.14 

0.020 

-1.12 

1.254 

0.28 

0.078 

0.00 

0.000 


2.62 

6.864 

0.73 

0.533 

0.06 

0.004 

-0.40 

0.160 

1.13 

1.277 



Each group of 25 or 50 deviates has been adjusted by a small amount so 
that the mean value of x for the group is exactly zero. The mean square 


deviate for each group is given at the extreme right. For a large enough group 
this average should approach unity. 


CONFIDENTIAL 


TABLES 


157 


Table IV. Binomial distribution function Fb(s, n) = 1 — I p (s + 1, n — s). Probability of s or fewer successes in n 
trials; and the probability that it will take more than n trials to achieve s + 1 successes. Probability of success per trial 
is p. (See equation (21) of Chapter 2.) 



II 

o 

i — * 

p = 0.2 

CO 

o 

II 

a 

p = 0A 

p = 0.5 

p = 0 . 6 

p = 0.7 

p = 0.8 

p = 0.9 

n = 2 , s = 0 

0.8100 

0.6400 

0.4900 

0.3600 

0.2500 

0.1600 

0.0900 

o'. 0400 

0.0100 

s = 1 

.9900 

.9600 

.9100 

.8400 

.7500 

.6400 

.5100 

.3600 

.1900 

s = 2 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1 . 0000 

1.0000 

n = 5, s = 0 

0.5905 

0.3277 

0.1681 

0.0778 

0.0512 

0.0102 

0.0024 

0.0003 

0.0000 

s = 1 

.9185 

.7373 

.5282 

.3370 

.1875 

.0949 

.0347 

.0067 

.0005 

s = 2 

.9914 

.9421 

.8369 

.6826 

.5000 

.3174 

.1631 

.0579 

.0086 

s =3 

.9995 

.9933 

.9692 

.9130 

.8125 

.6630 

.4718 

.2627 

.0815 

s = 4 

1.0000 

.9997 

.9976 

.9898 

.9688 

.9222 

.8319 

.6723 

.4095 

s = 5 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

n = 10 , s = 0 

0.3487 

0.1074 

0.0282 

0.0060 

0.0010 

0.0001 

0.0000 

0.0000 

0.0000 

s = 1 

.7361 

.4068 

.1493 

.0464 

.0107 

.0017 

.0001 

.0000 

.0000 

s =2 

.9298 

.6778 

.3828 

.1673 

.0547 

.0123 

.0016 

.0001 

.0000 

s = 3 

.9872 

.8791 

.6496 

.3823 

.1719 

.0548 

.0129 

.0012 

.0000 

s = 4 

.9990 

.9672 

.8497 

.6331 

.3770 

.1662 

.0463 

.0064 

.0001 

s = 5 

.9999 

.9936 

.9527 

. 8338 

.6230 

.3669 

.1503 

.0328 

.0016 

s = 6 

1.0000 

.9992 

.9894 

.9452 

.8281 

.6177 

.3504 

.1209 

.0128 

s = 7 

1.0000 

.9999 

.9984 

.9877 

.9453 

.8327 

.6172 

.3222 

.0702 

s = 8 

1.0000 

1.0000 

.9999 

.9983 

.9893 

.9536 

.8507 

.6242 

.2639 

s = 9 

1.0000 

1.0000 

1.0000 

.9999 

.9990 

.9940 

.9718 

.8926 

.6513 

s = 10 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

n = 20 , s = 0 

0.1216 

0.0115 

0.0008 

0.0000 

0.0000 

0.0000 

0.0000 

0.0000 

0.0000 

s = 2 

.6769 

.2161 

.0355 

.0036 

.0002 

.0000 

.0000 

.0000 

.0000 

s = 4 

.9568 

.6296 

.2375 

.0510 

.0059 

.0003 

.0000 

.0000 

.0000 

s = 6 

.9976 

.9133 

.6080 

.2500 

.0577 

.0064 

.0003 

.0000 

.0000 

s = 8 

.9999 

.9900 

.8867 

.5956 

.2517 

.0565 

.0051 

.0001 

.0000 

s = 10 

1.0000 

.9994 

.9829 

.8725 

.5881 

.2447 

.0480 

.0026 

.0000 

s = 12 

1.0000 

1.0000 

.9987 

.9790 

.8684 

.5841 

.2277 

.0321 

.0004 

s = 14 

1 . 0000 

1 . 0000 

1.0000 

.9984 

.9793 

.8744 

.5836 

.1958 

.0113 

s = 16 

1.0000 

1.0000 

1.0000 

1 . 0000 

.9987 

.9840 

.8929 

.5886 

.1330 

s = 18 

1.0000 

1 . 0000 

1.0000 

1.0000 

1.0000 

.9995 

.9924 

.9308 

.6083 

s = 20 

1.0000 

1.0000 

1.0000 

1.0000 

1 . 0000 

1.0000 

1.0000 

1.0000 

1 . 0000 

n = 50 , s = 0 

0.0052 

0.0000 

0.0000 

0.0000 

0.0000 

0.0000 

0.0000 

0.0000 

0.0000 

s = 5 

.6161 

.0480 

.0007 

.0000 

.0000 

.0000 

.0000 

.0000 

.0000 

s = 10 

.9906 

.5836 

.0789 

.0021 

.0000 

.0000 

.0000 

.0000 

.0000 

s = 15 

1.0000 

.9692 

.5692 

.0955 

.0033 

.0000 

.0000 

.0000 

.0000 

s = 20 

1 . 0000 

.9997 

.9522 

.5610 

.1013 

.0034 

.0000 

.0000 

.0000 

s = 25 

1.0000 

1.0000 

.9991 

.9427 

.5561 

.0978 

.0024 

.0000 

.0000 

s = 30 

1.0000 

1.0000 

1.0000 

.9986 

.9405 

.5535 

.0848 

.0009 

.0000 

s = 35 

1.0000 

1.0000 

1 . 0000 

1.0000 

.9987 

.9460 

.5532 

.0607 

.0001 

s = 40 

1.0000 

1 . 0000 

1 . 0000 

1.0000 

1.0000 

.9992 

.9598 

.5563 

.0245 

s = 45 

1.0000 

1.0000 

1 . 0000 

1.0000 

1 . 0000 

1.0000 

.9998 

.9815 

.5688 

s = 50 

1.0000 

1 . 0000 

1.0000 

1 . 0000 

1.0000 

1.0000 

1.0000 

1 . 0000 

1.0000 


CONFIDENTIAL 


158 


TABLES 


Table V . Normal distribution functions . 

/,(*) = -Le- (xV2 ); F.(x) = -+ C* e-WVdu. 

V 27 r V 27T c / — 00 

(See equation ( 24 ) of Chapter 2 .) 

' 

£ 

MX) 

F n (x) 

X 

fn(x) 

F n (x) 


- 4.0 

0.0001 

0.0000 

0.0 

0.3989 

0.5000 


- 3.8 

.0003 

.0001 

0.2 

.3910 

.5793 


- 3.6 

.0006 

.0002 

0.4 

.3683 

.6554 


- 3.4 

.0012 

.0004 

0.6 

.3332 

.7257 


- 3.2 

.0024 

.0007 

0.8 

.2897 

.7881 


- 3.0 

.0044 

.0014 

1.0 

.2420 

.8413 


— 2.8 

.0079 

.0026 

1.2 

.1942 

.8847 


- 2.6 

.0136 

.0047 

1.4 

.1497 

.9193 


- 2.4 

.0224 

.0082 

1.6 

.1109 

.9452 


- 2.2 

.0355 

.0139 

1.8 

.0790 

.9641 


- 2.0 

0.0540 

0.0228 

2.0 

0.0540 

0.9772 


- 1.8 

.0790 

.0359 

2.2 

.0355 

.9861 


- 1.6 

.1109 

.0548 

2.4 

.0224 

.9918 


- 1.4 

.1497 

.0807 

2.6 

.0136 

.9953 


- 1.2 

.1942 

.1151 

2.8 

.0079 

.9974 


- 1.0 

.2420 

.1587 

3.0 

.0044 

.9986 


- 0.8 

.2897 

.2119 

3.2 

.0024 

.9993 


- 0.6 

.3332 

.2743 

3.4 

.0012 

.9996 


- 0.4 

.3683 

.3446 

3.6 

.0006 

.9998 


- 0.2 

.3910 

.4207 

3.8 

.0003 

.9999 


0.0 

0.3989 

0.5000 

4.0 

0.0001 

1 . 0000 


Fnix) 

X 

fn(x) 

F nix) 

X 

fnix) 

Fnix) 

X 

fnix) 

0.00 

— OO 

0.0000 

0.30 

- 0.525 

0.3477 

0.70 

0.525 

0.3477 

.02 

- 2.053 

.0486 

.32 

- .468 

.3576 

.72 

.583 

.3363 

.04 

- 1.742 

.0874 

.34 

- .412 

.3665 

.74 

.644 

.3242 

.06 

- 1.554 

.1193 

.36 

- .359 

.3741 

.76 

.707 

.3109 

.08 

- 1.405 

.1489 

.38 

- .306 

.3807 

.78 

.773 

.2900 

.10 

- 1.281 

.1754 

.40 

- .253 

.3862 

.80 

.842 

.2799 

.12 

- 1.175 

0.1998 

.42 

- 0.202 

0.3906 

.82 

0.916 

0.2620 

.14 

- 1.080 

.2227 

.44 

- .151 

.3943 

.84 

0.995 

.2430 

.16 

- 0.995 

.2430 

.46 

- .101 

.3969 

.86 

1.080 

.2227 

.18 

- .916 

.2620 

.48 

- .050 

.3983 

.88 

1.175 

.1998 

.20 

- .842 

.2799 

.50 

0.000 

.3989 

.90 

1.281 

.1754 

.22 

- 0.773 

0.2900 

.52 

+ 0.050 

0.3983 

.92 

1.405 

0.1489 

.24 

- .707 

.3109 

.54 

.101 

.3969 

.94 

1.554 

.1193 

.26 

- .644 

.3242 

.56 

.151 

.3943 

.96 

1.742 

.0874 

.28 

- .583 

.3363 

.58 

.202 

.3906 

.98 

2.053 

.0486 

.30 

- .525 

.3477 

.60 

.253 

.3862 

1.00 

OO 

.0000 




.62 

0.306 

0.3807 







.64 

.359 

.3741 







.66 

.412 

.3665 







.68 

.468 

.3576 







.70 

.525 

.3477 





CONFIDENTIAL 


TABLES 


159 


E n / x m 

Table VI. Poisson distribution function F p (m, E) = L — : , e~ E = — - e~ x dx. Probability that m points or fewer are 

n=0 n\ ^ ml 

in an interval when the expected number is #. (See equation (30) of Chapter 2.) 


E = 0.1 

<N 

© 

II 

CO 

o 

II 

# = 0.4 

# = 0.5 

# = 0.6 

bd 

II 

© 

# = 0.8 

# = 0.9 


m = 0 

0.9048 

0.8187 

0.7408 

0.6703 

0.6065 

0.5488 

0.4966 

0.4493 

0.4066 


1 

.9953 

.9825 

.9631 

.9385 

.9098 

.8781 

.8442 

.8088 

.7725 


2 

.9998 

.9989 

.9964 

.9921 

.9856 

.9769 

.9659 

.9526 

.9371 


3 

1.0000 

.9999 

.9997 

.9992 

.9982 

.9966 

.9942 

.9909 

.9865 


4 

1 . 0000 

1.0000 

1.0000 

.9999 

.9998 

.9996 

.9992 

.9986 

.9977 


5 


1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

.9999 

.9998 

.9997 


6 




1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 


7 







1.0000 

1 . 0000 

1.0000 



E = 1.0 

# = 1.1 

# = 1.2 

# = 1.3 

# = 1.4 

# = 1.5 

# = 1.6 

# = 1.7 

# = 1.8 

# = 1.9 

n = 0 

0.3679 

0.3329 

0.3012 

0.2725 

0.2466 

0.2231 

0.2019 

0.1827 

0.1653 

0.1496 

1 

.7358 

.6990 

.6626 

.6268 

.5918 

.5578 

.5249 

.4932 

.4628 

.4337 

2 

.9197 

.9004 

.8795 

.8571 

.8335 

.8088 

.7834 

.7572 

.7306 

.7037 

3 

.9810 

.9743 

.9662 

.9569 

.9463 

.9344 

.9212 

.9068 

.8913 

.8747 

4 

.9963 

.9946 

.9923 

.9893 

.9857 

.9814 

.9763 

.9704 

.9636 

.9559 

5 

.9994 

.9990 

.9985 

.9178 

.9968 

.9955 

.9940 

.9920 

.9896 

.9868 

6 

.9999 

.9999 

.9997 

.9996 

.9994 

.9991 

.9987 

.9981 

.9974 

.9966 

7 

1.0000 

1.0000 

1.0000 

.9999 

.9999 

.9998 

.9997 

.9996 

.9994 

.9992 

8 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1 . 0000 

1.0000 

.9999 

.9999 

.9998 

9 




1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1 . 0000 


# = 2.0 

# = 2.1 

# = 2.2 

# = 2.3 

# = 2.4 

# = 2.5 

# = 2.6 

# = 2.7 

# = 2.8 

# = 2.9 

m = 0 

0.1353 

0.1225 

0.1108 

0.1003 

0.0907 

0.0821 

0.0743 

0.0672 

0.0608 

0.0550 

1 

. 4060 

.3796 

.3546 

.3309 

.3084 

.2873 

.2674 

.2487 

.2311 

.2146 

2 

.6767 

.6496 

.6227 

.5960 

.5697 

.5438 

.5184 

.4936 

.4695 

.4460 

3 

.8571 

.8386 

.8194 

.7993 

.7787 

.7576 

.7360 

.7141 

.6919 

.6696 

4 

.9473 

.9379 

.9275 

.9102 

.9041 

.8912 

.8774 

.8629 

.8477 

.8318 

5 

.9834 

.9796 

.9751 

.9700 

.9643 

.9580 

.9510 

.9433 

.9349 

.9258 

6 

.9955 

.9941 

.9925 

.9906 

.9884 

.9858 

.9828 

.9794 

.9756 

.9713 

7 

.9989 

.9985 

.9980 

.9974 

.9967 

.9958 

.9947 

.9934 

.9919 

.9912 

8 

.9998 

.9997 

.9995 

.9994 

.9991 

.9989 

.9985 

.9981 

.9976 

.9969 

9 

1 . 0000 

.9999 

.9999 

.9999 

.9998 

.9997 

.9996 

.9995 

.9993 

.9991 

10 

1.0000 

1.0000 

1.0000 

1.0000 

1 . 0000 

.9999 

.9999 

.9999 

.9998 

.9998 

12 


1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

14 






1 . 0000 

1.0000 

1.0000 

1.0000 

1.0000 


# = 3.0 

# = 3.2 

II 

CO 

4^ 

# = 3.6 

ba 

ll 

CO 

bo 

# = 4.0 

# = 4.2 

TjH 

TjH 

II 

K? 

CD 

II 

oo 

II 

m = 0 

0.0498 

0.0408 

0.0334 

0.0273 

0.0224 

0.0183 

0.0150 

0.0123 

0.0101 

0.0082 

1 

.1991 

.1712 

.1468 

.1257 

.1074 

.0916 

.0780 

.0663 

.0563 

.0477 

2 

.4232 

.3799 

.3397 

.3027 

.2689 

.2381 

.2102 

.1851 

.1626 

.1425 

3 

.6472 

.6025 

.5584 

.5152 

.4735 

.4335 

.3954 

.3594 

.3257 

.2942 

4 

.8153 

.7806 

.7442 

.7064 

.6678 

.6288 

.5898 

.5512 

.5132 

.4763 

5 

.9161 

.8946 

.8705 

.8441 

.8156 

.7851 

.7531 

.7199 

.6858 

.6510 

6 

.9665 

.9554 

.9421 

.9267 

.9091 

.8893 

.8675 

.8436 

.8180 

.7908 

7 

.9881 

.9832 

.9769 

.9692 

.9599 

.9489 

.9361 

.9214 

.9050 

.8867 

8 

.9962 

.9943 

.9917 

.9883 

.9840 

.9786 

.9721 

.9642 

.9549 

.9442 

9 

.9989 

.9982 

.9973 

.9960 

.9942 

.9919 

.9889 

.9851 

.9805 

.9749 

10 

.9997 

.9995 

.9992 

.9987 

.9981 

.9972 

.9959 

.9943 

.9922 

.9896 

12 

1 . 0000 

1 . 0000 

.9999 

.9999 

.9998 

.9997 

.9996 

.9993 

.9990 

.9986 

14 

1.0000 

1.0000 

1.0000 

1 . 0000 

1.0000 

1.0000 

1.0000 

.9999 

.9999 

.9999 

16 



1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

18 








1 . 0000 

1 . 0000 

1.0000 













CONFIDENTIAL 


160 


TABLES 


Table VI — ( Continued ) 


£ = 5.0 

£ = 5.2 

£ = 5.4 

£ = 5.6 

£ = 5.8 

£ = 6.0 

£ = 6.5 

£ = 7.0 

£ = 7.5 

£ = 8.0 


0 

0.0067 

0.0055 

0.0045 

0.0037 

0.0030 

0.0025 

0.0015 

0.0009 

0.0006 

0.0003 

1 

.0404 

.0342 

.0289 

.0244 

.0206 

.0174 

.0113 

.0073 

.0047 

.0030 

2 

.1247 

.1088 

.0948 

.0824 

.0715 

.0620 

.0430 

.0296 

.0203 

.0137 

3 

.2650 

.2381 

.2133 

.1906 

.1700 

.1512 

.1118 

.0818 

.0591 

.0424 

4 

.4405 

.4061 

.3733 

.3422 

.3127 

.2851 

.2237 

.1730 

.1321 

.0996 

5 

.6160 

.5809 

.5461 

.5119 

.4783 

.4457 

.3690 

.3007 

.2414 

.1912 

6 

.7622 

.7324 

.7017 

.6703 

.6384 

.6063 

.5265 

.4497 

.3782 

.3134 

7 

.8666 

.8449 

.8217 

.7970 

.7710 

.7440 

.6728 

.5987 

.5246 

.4529 

8 

.9319 

.9181 

.9027 

.8857 

.8672 

.8472 

.7916 

.7291 

.6620 

.5925 

9 

.9682 

.9603 

.9512 

.9409 

.9292 

.9161 

.8774 

.8305 

.7764 

.7166 

10 

' .9863 

.9823 

.9775 

.9718 

.9651 

.9574 

.9332 

.9015 

.8622 

.8159 

12 

.9980 

.9972 

.9962 

.9949 

.9932 

.9912 

.9840 

.9730 

.9573 

.9362 

14 

.9998 

.9997 

.9995 

.9993 

.9990 

.9986 

.9970 

.9943 

.9897 

.9827 

16 

1.0000 

1.0000 

.9999 

.9999 

.9999 

.9998 

.9996 

.9990 

.9980 

.9963 

18 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

.9999 

.9999 

.9997 

.9993 

20 



1.0000 

1.0000 

1.0000 

1 . 0000 

1 . 0000 

1.0000 

1.0000 

.9999 

22 







1 . 0000 

1 . 0000 

1 . 0000 

1 . 0000 

24 










1.0000 


£ = 8.5 


m = 0 0.0002 

1 .0019 

2 .0093 

3 .0301 

4 . 0744 

5 . 1496 

6 . 2562 

7 .3856 

8 .5231 

9 . 6530 


£ = 9.0 E = 9.5 


0.0001 

0.0001 

.0012 

.0008 

.0062 

.0042 

.0212 

.0149 

.0550 

.0403 

.1157 

.0885 

.2068 

.1650 

.3239 

.2687 

.4557 

.3918 

.5874 

.5218 


£ = 10 


0.0000 

m = 10 

.0005 

12 

.0028 

14 

.0103 

16 

.0293 

18 

.0671 

20 

.1301 

22 

.2202 

24 

.3328 

26 

.4579 



£ = 8.5 £ = 9.0 


.7634 

.7060 

.9091 

.8758 

.9726 

.9585 

.9934 

.9889 

.9987 

.9976 

.9998 

.9996 

1.0000 

.9999 

1.0000 

1.0000 
1 . 0000 


£ = 9.5 £ = 10 


.6453 

.5830 

.8364 

.7916 

.9400 

.9165 

.9823 

.9730 

.9957 

.9928 

.9991 

.9984 

.9999 

.9997 

1 . 0000 

1.0000 

1.0000 

1 . 0000 


CONFIDENTIAL 


BIBLIOGRAPHY 


1. U. S. Submarine Current Doctrine, U. S. F. 25- A, Issuing 
Office, U. S. Navy Department. 

2. The Causes of Evolution, J. B. S. Haldane, Harper and 
Brothers, London, 1934. 

3. The Genetical Theory of Natural Selection, R. A. Fisher, 
Clarendon Press, Oxford, 1930. 

4. Theory of Games and Economic Behavior, John Von Neu- 
mann and Oskar Morgenstern, Princeton University Press, 
Princeton, 1944. 

5. Probability and Its Engineering Uses, Thornton C. Fry, 
D. Van Nostrand and Company, Inc., New York, 1928. 

6. Tables of the Incomplete Beta-Function, Karl Pearson, edi- 
tor, University College, London, 1934. 

7. Handbook of Chemistry and Physics, Charles D. Hodgman, 
editor, Chemical Rubber Publishing Co., Cleveland, 29th 
edition, 1945. 

8. Handbook of Mathematical Tables and Formulas, compiled 
by Richard Stevens Burington, Handbook Publishers, Inc., 
Sandusky, Ohio, 2nd edition, 1940. 


9. Aircraft in Warfare ; the Dawn of the Fourth Arm, F. W. 
Lanchester, Constable and Company, Ltd., London, 1916. 

10. A Quantitative Aspect of Combat, B. 0. Koopman, OEMsr- 

1007, Applied Mathematics Panel, Note 6, AMG-Colum- 
bia, August 1943. AMP-900-M2* 

1 1 . Legons sur la theorie mathematique de la lutte pour la vie, Vito 
Volterra, Gauthier- Villars et cie., Paris, 1931. 

12. Elements of Physical Biology, Alfred J. Lotka, Williams and 
Wilkins Company, Baltimore, 1925. 

13. A Treatise on the Theory of Bessel Functions, G. N. Watson, 
University Press, Cambridge, 2nd edition, 1944. 

14. Random Sampling Numbers, arranged by L. H. C. Tippett, 
Cambridge University Press, London, 1927. 

15. Bollettino di Matematica, E. L. Dodd, 1942, pp. 76-77. 


* This number indicates that this document has been microfilmed and 
that its title appears in the microfilm index printed in a separate volume. 
For access to the index volume and to the microfilm, consult the Army or 
Navy agency listed on the reverse of the half-title page. 


CONFIDENTIAL 


161 


CONTRACT NUMBERS, CONTRACTORS, AND SUBJECTS OF CONTRACTS 


Contract 

Number 

Name and Address 
of Contractor 

Subject 

OEMsr-20 

The Trustees of Columbia University 
in the City of New York 

New York, N. Y. 

Studies and experimental investigations in connection with 
and for the development of equipment and methods per- 
taining to submarine warfare. 

OEMsr-1128 

The Trustees of Columbia University 
in the City of New York 

New York, N. Y. 

Conduct studies and experimental investigations in connec- 
tion with and for the development of equipment and 
methods involved in submarine and subsurface warfare. 


162 


CONFIDENTIAL 


SERVICE PROJECT NUMBERS 


The projects listed below were transmitted to the Executive Secre- 
tary, National Defense Research Committee [NDRC], from the War 
or Navy Department through either the War Department Liaison 
Officer for NDRC or the Office of Research and Inventions (formerly 
the Coordinator of Research and Development), Navy Department. 


Service Project Number 

Title 

AC-50 

Operations Research 

NR-100 

Operations Research 


CONFIDENTIAL 


163 





INDEX 


The subject indexes of all STR volumes are combined in a master index printed in a separate volume. 

For access to the index volume consult the Army or Navy Agency listed on the reverse of the half-title page. 


Accuracy measurements for weapons, 
132-134 

Accuracy of bombing, 58 
Acoustic torpedo countermeasures, 
94-95 

Air escort requirements for convoys, 
62-63 

Aircraft antisubmarine depth bombs, 
55-56 

Aircraft disposition in Bay of Biscay, 
49-50 

Aircraft load-carrying improvement, 58 
Aircraft requirements for antisub- 
marine protection, 61-62 
Aircraft search for submarines, 40-45 
Bay of Biscay antisubmarine flying, 
43-45 

distribution of flying effort, 42-43 
Aircraft versus submarine as anti-ship 
weapons, 50-51 

Air-to-air combat exchange rates, 45-46 
Alphabetic accounting machine, 151 
Analytical methods for operations re- 
search, 6 

Antiaircraft fire against suicide planes, 
81-84 

Antiaircraft gunnery, evaluation, 135- 
136 

Antiaircraft gunnery, hit expectancy 
curve, 136 

Antiaircraft guns for merchant vessels, 
evaluation, 52-53 
Antiaircraft splash power, 87-88 
Anti-ship weapons, 48-51 
Antisubmarine barrier patrol tactics, 
105-109 

safe tactics for barrier patrol, 108 
safe tactics for both sides, 108 
safe tactics for submarine, 107-108 
Antisubmarine flying in Bay of Biscay, 
43-45 

British S-band radar, 45 
effectiveness of night flying, 43 
German radar receiver success, 43 
Antisubmarine protection, aircraft re- 
quirements, 61-62 
Antitorpedo nets, 53 
Area bombardment, repairability prob- 
ability, 87 

Ballistic errors, standard deviation cal- 
culations, 132-134 

Barrier patrol tactics, antisubmarine, 
105-109 


safe tactics for barrier patrol, 108 
safe tactics for both sides, 108 
safe tactics for submarine, 107- 
108 

Binomial distribution, 22-26 
Binomial distribution function, 157 
Bombardment problems, 110-128 
destructiveness of weapons, lethal 
area, 110-113 

pattern firing, ballistic dispersion 
present, 117-122 

pattern firing, no ballistic dispersion, 
113-117 

sampling method, 122-128 
Bombing accuracy evaluation, 58 
Bombing calculation by sampling 
method, 125-128 

Bombing training evaluation, 57-58 
Bombsights, radar, 56-57 
Borkum search receiver, 96 
British, S-band radar, 45 
Buffon’s needle problem, 15-16 

Card-counting sorter, 150 
Chi-squared test, probability theory, 
33-36 

Circular normal law, 36-38 
Combat air patrol distribution about 
task forces, 89-92 

analysis of tactical situation, 90-91 
multiple units, 91-92 
Combat exchange rates, air-to-air, 
45-46 

Comparative effectiveness 

anti-ship versus anti-city bombing, 
48-51 

anti-ship weapons, 48-49 
bombing U-boat pens versus escort- 
ing convoys, 49-50 
submarine versus aircraft as anti- 
ship weapons, 50-51 
Compound probabilities, 16-19 
area bombing as example, 17-19 
conditional probabilities, 16-17 
Convoy versus submarine exchange 
rates, 46-48 

Countermeasure action, theoretical 
analysis, 102-109 

barrier patrol for submarines, 105- 
109 

case of three choices, 105 
definite case, 103 
indefinite case, 103-105 
minimax principle, 103 


Countermeasures, 94-109 

continuous operations, 101-102 
discrete operational trials, 98-99 
effectiveness of enemy’s measures, 
98-99 

rules for operational trials, 100-101 
success probability, mathematics of, 
99 

to acoustic torpedoes, 94-95 
to radar, 95-98 

Data collecting and recording, statis- 
tical methods, 4-5, 146 
Depth bombs, aircraft antisubmarine, 
55-56 

Depth charge expenditure, 62-63 
Destruction of production, 73-74 
Destructiveness of weapons, lethal 
area, 110-113 

aimed fire, small targets, 112-113 
damage coefficients, 1 10-1 1 1 
multiple hits, 110-111 
random or area bombardment, 111- 
112 

Detection equipment evaluation, 134- 
135 

detection expectancy, 135 
range distribution method, 135 
sweep width, 134 

Distribution functions, probability 
theory, 12-16 
in several variables, 15-16 
probability density, 13-16 
random variable, 12-14 
search as example, 13 
stochastic variable, 12-16 
Distribution laws, 22-32 
binomial distribution, 22-26 
normal distribution, 26-28 
Poisson distribution 28-32 

Effective comparisons 

anti-ship versus anti-city bombing, 
48-51 

anti-ship weapons, 48-49 
bombing U-boat pens versus escort- 
ing convoys, 49-50 
submarine versus aircraft as anti-ship 
weapons, 50-51 

Electric torpedo evaluation, 54-55 
Enemy countermeasures, effectiveness, 
59-60, 98-102 

continuous operations, 101-102 
discrete operational trials, 98-99 


CONFIDENTIAL 


165 


166 


INDEX 


evaluation, 59-60 

rules for operational trials, 100-101 
success probability, mathematical 
determination, 99 
Equations, Lanchester’s 
see Lanchester’s equations 
Equations for flow of submarines, 
78-80 

Equipment performance evaluation, 
51-60 

aircraft antisubmarine depth bombs, 
55-56 

antiaircraft guns for merchant ves- 
sels, 52-53 
antitorpedo nets, 53 
devising operational practice train- 
ing, 52 

effects of training, 57-59 
enemy countermeasure evaluation, 
59-60 

evaluation questionnaire, 52 
height-finding radar, 56 
magnetic airborne detector, 53-54 
maintenance importance, 56 
new equipment, first use, 51-52 
radar bombsights, 56-57 
submarine torpedo, 54-55 
Exchange rates 

air-to-air combat, 45-46 
convoy versus submarine, 45-48 
Experiment planning, operational, 129- 
132 

Conclusions, 131-132 
plan of procedure, 130-131 
preliminary theory, 130 
preliminary write-up, 130 
significant data, 131 
“Expert opinion,” limitations, 5-6 

Field activities, 140-144 
assignment types, 140-142 
general recommendations, 143-144 
work types, 142-143 
Fighter aircraft, Japanese versus 
American, 45-46 

Force requirement determination, 61- 
63 

air escort requirements, 62-63 
depth charge expenditure, 62-63 

German U-boat search receivers, 95-97 
Borkum receiver, 96 
Metox receiver, 95 
Naxos receiver, 96-97 
Tunis receiver, 97 
Wanz G1 receiver, 96 
Gunnery, survival problems, 135-136 
Gunnery and bombardment problems, 
110-128 

destructiveness of weapons, lethal 
area, 110-113 


pattern firing, ballistic dispersion 
present, 117-122 

pattern firing, no ballistic dispersion, 
113-117 

sampling method, 122-128 

Height-finding radar, 56 
Historic background for operations re- 
search, 1 

Hit expectancy curve for AA gunnery, 
136 

IBM (Hollerith) cards, 146-148 
IBM (Hollerith) machines, 150-151 
alphabetic accounting machine, 151 
alphabetic interpreter, 150 
alphabetic key punch, 149 
alphabetic verifier, 150 
card-counting sorter, 150 
collator, 151 

Japanese submarines, operational data, 
85-86 

Japanese suicide plane problem, 81-84 
damage due to suicide planes, 81-82 
effect of angle of approach, 82-84 
ship maneuvering, 82 
suggested ship tactics, 84 
Japanese versus American fighter air- 
craft, 45-46 

Key sort cards, 148-149 
Kinematics, strategic, 61-80 
force requirements, 61-63 
Lanchester’s equations, 63-77 
reaction rate problems, 77-80 

Lanchester’s equations, 63-77 
concentration of resources, 64 
destruction of production, 73-74 
fighting strength, 66 
generalized equations, 71-77 
linear law, 65, 67-69 
loss rates, 71-72 
minimax principle, 75-77 
probability analysis, 67-71 
square law, 65-67, 69 
tactical and strategic forces, 74-75 
typical solutions, 72-73 
Lethal area of weapons, 1 10-1 1 3 
aimed fire, small targets, 112-113 
damage coefficients, 1 10-1 1 1 
multiple hits, 1 10-1 1 1 
random or area bombardment, 111- 
112 

Limitations of “expert opinion,” 5-6 
Limitations of operational data, 5 
Linear law, Lanchester, 65, 67-69 

Magnetic airborne detector evaluation, 
53-54 


Mark 18 torpedo evaluation, 54-55 
Measures of effectiveness for operations 
research, 38-60 

comparative effectiveness, 48-51 
evaluation of equipment perform- 
ance, 51-60 
exchange rates, 45-48 
“hemibel thinking,” 38 
sweep rates, 38-45 
Metox search receiver, 95 
Mine field effectiveness study, 87 
Minimax principle, 75-77, 103 

Naxos search receiver, 96-97 
Noisemaker, FXR, 95 
Normal distribution functions, 158 
Normal distribution, probability 
theory, 26-28 

carrier bombing example, 28 
limiting curve equation, 27 
normal distribution curve, 26-27 

Operational experiment planning, 129- 
132 

conclusions, 131-132 
plan of procedure, 130-131 
preliminary theory, 130 
preliminary write-up, 130 
significant data, 131 
Operational experiments with equip- 
ment and tactics, 6, 129-136 
accuracy measurements, 132-134 
antiaircraft gunnery evaluation, 135- 
136 

detection equipment evaluation, 
134-135 

planning of experiments, 129-132 
survival problems in gunnery, 135- 
136 

Organizational and procedural prob- 
plems in operations research, 
137-151 

assignment of group, 138 
field activities, 140-144 
importance of contacts with several 
• echelons, 137-138 
recruiting and training operations re- 
search workers, 140 
reports and memoranda, 139 
statistical methods, 144-151 
status of group members, 140 
sub-groups in field, 138-139 

Pattern firing, ballistic dispersion 
present, 117-122 

approximate method, example, 120- 
122 

approximate solution for large pat- 
terns, 118-120 

pattern damage function, 1 17— 

118 


CONFIDENTIAL 


INDEX 


167 


pattern density function, 118-120 
probability estimates by approxi- 
mate method, 121-122 
Pattern firing, no ballistic dispersion, 
113-117 

pattern damage function, 1 14 
squid problem, 116-117 
train bombing example, 114-116 
Pattern firing problems, sampling 
method, 122-128 

construction of sampling popula- 
tions, 122-123 
rocket problem, 123-124 
short cuts, 124-125 
train bombing, 125-128 
Peacetime applications of operations re- 
search, 6-10 

administrator-scientist relationship , 
8-9 

analytical study of operational con- 
stants, 8 

constancy of operational constants, 
7-8 

examples of applications, 9-10 
measures of value, 8 
Poisson distribution, 28-32, 88 
application to aerial search, 30-31 
area bombardment, 87 
equation, 29-30 
hit probability, 86, 119, 136 
newsboy example, 31-32 
Poisson distribution function, 30, 159- 
160 

Probability analysis of Lanchester’s 
equations, 67-71 
example, 70-71 
linear law, 67-69 
square law, 69 

Probability calculations, train bomb- 
ing, 125-128 

Probability density, 13-16 
Probability estimates by approximate 
method, 121-122 
Probability of hit, 86-87 
Probability theory, distribution func- 
tions, 12-16 

Probability theory, distribution laws, 
22-32 

binomial distribution, 22-26 
normal distribution, 26-28 
Poisson distribution, 28-32 
Probability theory, fundamental con- 
cepts, 11-21 

compound probabilities, 16-19 
distribution functions, 12-16 
expected values, 19-21 
probability element, 1 1-12 
randomness, 11 
standard deviation, 20-21 
Probability theory, sampling, 32-37 
chi-squared test, 33-36 


circular normal law, 36-38 
example tests, 36-38 
“reasonableness/’ 32-33 
Procedural problems 
see Organizational and procedural 
problems in operations research 
Punch card system, 146-148 
code sample, 147-148 
IBM (Hollerith) cards, 146-148 
kinds of cards, 145-146 
objections, 145-146, 148 

Radar, height-finding, 56 
Radar bombsights, 56-57 
Radar countermeasures, 95-98 
Allied S-band radar, 96 
Allied X-band equipment, 97 
German Borkum receiver, 96 
German Metox receiver, 95 
German Naxos search receiver, 96-97 
German Tunis receiver, 97 
German Wanz G1 receiver, 96 
intermittent operation, 97-98 
Vixen attenuators, 96 
Random normal deviates in units of 
standard errors, 156 
Random sequence of angles, 155 
Random sequence of digits, 153-154 
Random variable in probability theory, 
12-14 

Randomness, 11 
Reaction rate problems, 77-80 
circulation of U-boats, 77-78 
equations of flow, 77-78 
typical solutions, 79-80 
Recruiting and training operations re- 
search workers, 140 
biologists, 140 
chemists, 140 
mathematicians, 140 
physicists, 140 
special courses, 140 

Sampling method for pattern problems, 
122-128 

construction of sampling popula- 
tions, 122-123 
rocket problem, 123-124 
shortcuts, 124-125 
train bombing, 125-128 
Science in warfare, increasing impor- 
tance, 1-3 

Scope of operations research 1-4 
Search receivers, German, 95-97 
Borkum receiver, 96 
Metox receiver, 95 
Naxos receiver, 96-97 
Tunis receiver, 97 
Wanz G1 receiver, 96 
Search theory, analytical solutions, 
86-94 


antiaircraft splash power, 87-88 
area bombardment, 87 
covered area, 86 

disposition of CAP protection about 
task forces, 89-92 
merchant vessel sinkings, 87 
probability of hit, 86-87 
submarine effectiveness analysis, 88- 
89 

tactics to evade torpedoes, 92-94 
Ship maneuvering to dodge suicide 
planes, 81-84 

Square law, Lanchester, 65-67, 69 
Squid effectiveness determination, 116— 
117 

Standard deviation calculations for bal- 
listic errors, 132-134 
Standard deviation of stochastic var- 
iable, 20-21 

Statistical methods in operations re- 
search, 4, 144-151 
general purposes, 144-145 
IBM (Hollerith) cards, 146-148 
key sort cards, 148-149 
ledger and dual purpose cards, 148 
mechanics of analysis, 149 
mechanics of collecting and recording 
data, 146 

planning procedure, 145-146 
preparation of final report, 151 
punch card system, 145-146 
study of data, 151 

Statistical solutions to tactical prob- 
lems, 81-86 

approach angle of diving plane, 82-84 
damage due to suicide planes, 81-82 
effects of ship maneuvering, 82 
submarine casualties, 84-86 
suggested ship tactics, 84 
Steam torpedo evaluation, 54-55 
Stochastic variable in probability 
theory, 12-16 

Strategic kinematics, 61-80 
force requirements, 61-63 
generalized Lanchester equations, 

71-77 

Lanchester’s equations, 63-67 
probability analysis of Lanchester’s 
equations, 67-71 
reaction rate problems, 77-80 
Submarine casualty causes, 84-86 
comparison with Japanese casualties, 
84-85 

operational data, 85 
suggested measures, 86 
Submarine circulation, World War II, 
77-78 

Submarine effectiveness analysis, 88-89 
group operation, 89 
independent patrol, 88 
Submarine flow equations, 78-80 


CONFIDENTIAL 


168 


INDEX 


Submarine patrol, 40 
Submarine search by aircraft, 40-45 
Submarine torpedo evaluation, 54-55 
Submarine versus aircraft as anti-ship 
weapon, 50-51 

Suicide planes, antiaircraft fire against, 
81-84 

Sweep rates, 38-45 

aircraft search for submarines, 40-45 
antisubmarine flying in Bay of Bis- 
cay, 43-45 

calculation of constants, 39 
distribution of flying effort, 42-43 
operational sweep rate, 39 
submarine patrol, 40 
theoretical sweep rate, 39-40 


Sweep width, 134 

Tactical analysis by operations research 
methods, 81-109 

measure and countermeasure, 94-102 

solutions involving search theory, 
86-94 

statistical solutions, 81-86 
Target hit probability, 86-87 
Theory of search 

see Search theory, analytical solu- 
tions 

Torpedo evaluation, submarine, 54- 
55 

Torpedo evasion tactics, 92-94 
Train bombing, probability calcula- 
tions, 125-128 


/ DECLASSIFIED 
By authority Secretary of 

SEP 7 1960 

Defense memo 2 August 1960 
XJBRARY OF CONGRESS 



Training operations research workers, 
140 

Tunis search receiver, 97 

U-boat circulation, World War II, 77- 
78 

Wanz G1 search receiver, 96 
Weapon accuracy measurements, 132- 
134 

Weapon destructiveness, lethal area, 
110-113 

aimed fire, small targets, 112-113 
damage coefficients, 1 10-1 1 1 
multiple hits, 1 10-1 1 1 
random or area bombardment, 111- 
112 




r 















